diff --git a/mbcp/mp_math/angle.py b/mbcp/mp_math/angle.py
index f339c06..6240b96 100644
--- a/mbcp/mp_math/angle.py
+++ b/mbcp/mp_math/angle.py
@@ -8,6 +8,7 @@ Copyright (C) 2020-2024 LiteyukiStudio. All Rights Reserved
 @File    : angle.py
 @Software: PyCharm
 """
+import math
 from typing import overload
 
 from .const import PI  # type: ignore
@@ -71,6 +72,60 @@ class AnyAngle:
         """
         return AnyAngle(-self.radian % (2 * PI), is_radian=True)
 
+    @property
+    def sin(self) -> float:
+        """
+        正弦值。
+        Returns:
+            正弦值
+        """
+        return math.sin(self.radian)
+
+    @property
+    def cos(self) -> float:
+        """
+        余弦值。
+        Returns:
+            余弦值
+        """
+        return math.cos(self.radian)
+
+    @property
+    def tan(self) -> float:
+        """
+        正切值。
+        Returns:
+            正切值
+        """
+        return math.tan(self.radian)
+
+    @property
+    def cot(self) -> float:
+        """
+        余切值。
+        Returns:
+            余切值
+        """
+        return 1 / math.tan(self.radian)
+
+    @property
+    def sec(self) -> float:
+        """
+        正割值。
+        Returns:
+            正割值
+        """
+        return 1 / math.cos(self.radian)
+
+    @property
+    def csc(self) -> float:
+        """
+        余割值。
+        Returns:
+            余割值
+        """
+        return 1 / math.sin(self.radian)
+
     def __add__(self, other: 'AnyAngle') -> 'AnyAngle':
         return AnyAngle(self.radian + other.radian, is_radian=True)
 
diff --git a/mbcp/mp_math/const.py b/mbcp/mp_math/const.py
index d7bc4b9..ac54de6 100644
--- a/mbcp/mp_math/const.py
+++ b/mbcp/mp_math/const.py
@@ -16,4 +16,6 @@ E = math.e
 GOLDEN_RATIO = (1 + math.sqrt(5)) / 2
 GAMMA = 0.57721566490153286060651209008240243104215933593992
 EPSILON = 0.0001
-
+"""ε"""
+APPROX = 0.001
+"""约等于误差"""
diff --git a/mbcp/mp_math/line.py b/mbcp/mp_math/line.py
index 93945ac..9fdde2d 100644
--- a/mbcp/mp_math/line.py
+++ b/mbcp/mp_math/line.py
@@ -8,8 +8,10 @@ Copyright (C) 2020-2024 LiteyukiStudio. All Rights Reserved
 @File    : other.py
 @Software: PyCharm
 """
+import math
 from typing import TYPE_CHECKING
 
+from .const import APPROX
 from .mp_math_typing import OneSingleVarFunc, RealNumber
 from .utils import sign_format
 from .vector import Vector3
@@ -30,6 +32,17 @@ class Line3:
         self.point = point
         self.direction = direction
 
+    def approx(self, other: 'Line3', epsilon: float = APPROX) -> bool:
+        """
+        判断两条直线是否近似相等。
+        Args:
+            other: 另一条直线
+            epsilon: 误差
+        Returns:
+            是否近似相等
+        """
+        return self.is_approx_parallel(other, epsilon) and (self.point - other.point).is_approx_parallel(self.direction, epsilon)
+
     def cal_angle(self, other: 'Line3') -> 'AnyAngle':
         """
         计算直线和直线之间的夹角。
@@ -42,6 +55,28 @@ class Line3:
         """
         return self.direction.cal_angle(other.direction)
 
+    def cal_distance(self, other: 'Line3 | Point3') -> float:
+        """
+        计算直线和直线或点之间的距离。
+        Args:
+            other: 平行直线或点
+
+        Returns:
+            距离
+        Raises:
+            ValueError: 直线不平行
+            TypeError: 不支持的类型
+        """
+        if isinstance(other, Line3):
+            if self.is_parallel(other):
+                return (self.point - other.point).cross(self.direction).length / self.direction.length
+            else:
+                raise ValueError("Lines are not parallel.")
+        elif isinstance(other, Point3):
+            return (other - self.point).cross(self.direction).length / self.direction.length
+        else:
+            raise TypeError("Unsupported type.")
+
     def cal_intersection(self, other: 'Line3') -> 'Point3':
         """
         计算两条直线的交点。
@@ -49,12 +84,16 @@ class Line3:
             other: 另一条直线
         Returns:
             交点
+        Raises:
+            ValueError: 直线平行
+            ValueError: 直线不共面
         """
         if self.is_parallel(other):
             raise ValueError("Lines are parallel and do not intersect.")
         if not self.is_coplanar(other):
             raise ValueError("Lines are not coplanar and do not intersect.")
-        return self.point + self.direction.cross(other.direction)
+        return (self.point + (self.direction.cross(other.direction) @ other.direction.cross(self.point - other.point)) /
+                self.direction.cross(other.direction).length ** 2 * self.direction)
 
     def cal_perpendicular(self, point: 'Point3') -> 'Line3':
         """
@@ -86,6 +125,17 @@ class Line3:
                 lambda t: self.point.y + self.direction.y * t,
                 lambda t: self.point.z + self.direction.z * t)
 
+    def is_approx_parallel(self, other: 'Line3', epsilon: float = 1e-6) -> bool:
+        """
+        判断两条直线是否近似平行。
+        Args:
+            other: 另一条直线
+            epsilon: 误差
+        Returns:
+            是否近似平行
+        """
+        return self.direction.is_approx_parallel(other.direction, epsilon)
+
     def is_parallel(self, other: 'Line3') -> bool:
         """
         判断两条直线是否平行。
@@ -106,15 +156,26 @@ class Line3:
         """
         return self.is_parallel(other) and (self.point - other.point).is_parallel(self.direction)
 
+    def is_point_on(self, point: 'Point3') -> bool:
+        """
+        判断点是否在直线上。
+        Args:
+            point: 点
+        Returns:
+            是否在直线上
+        """
+        return (point - self.point).is_parallel(self.direction)
+
     def is_coplanar(self, other: 'Line3') -> bool:
         """
         判断两条直线是否共面。
+        充要条件：两直线方向向量的叉乘与两直线上任意一点的向量的点积为0。
         Args:
             other: 另一条直线
         Returns:
             是否共面
         """
-        return self.direction.cross(other.direction).is_parallel(self.direction)
+        return self.direction.cross(other.direction) @ (self.point - other.point) == 0
 
     def simplify(self):
         """
@@ -147,15 +208,20 @@ class Line3:
         direction = p2 - p1
         return cls(p1, direction)
 
-    def __and__(self, other: 'Line3') -> 'Point3':
+    def __and__(self, other: 'Line3') -> 'Line3 | Point3 | None':
         """
-        计算两条直线点集合的交集。交点
+        计算两条直线点集合的交集。重合线返回自身，平行线返回None，交线返回交点。
         Args:
             other: 另一条直线
         Returns:
             交点
         """
-        return self.cal_intersection(other)
+        if self.is_collinear(other):
+            return self
+        elif self.is_parallel(other) or not self.is_coplanar(other):
+            return None
+        else:
+            return self.cal_intersection(other)
 
     def __eq__(self, other) -> bool:
         """
diff --git a/mbcp/mp_math/plane.py b/mbcp/mp_math/plane.py
index a979e62..e87deff 100644
--- a/mbcp/mp_math/plane.py
+++ b/mbcp/mp_math/plane.py
@@ -7,10 +7,11 @@ from typing import TYPE_CHECKING, overload
 
 import numpy as np
 
+from .const import APPROX
 from .vector import Vector3, zero_vector3
 from .line import Line3
 from .point import Point3
-from .utils import sign
+from .utils import approx, sign
 
 if TYPE_CHECKING:
     from .angle import AnyAngle
@@ -31,6 +32,28 @@ class Plane3:
         self.c = c
         self.d = d
 
+    def approx(self, other: 'Plane3', epsilon: float = APPROX) -> bool:
+        """
+        判断两个平面是否近似相等。
+        Args:
+            other:
+            epsilon:
+
+        Returns:
+            是否近似相等
+        """
+        if self.a != 0:
+            k = other.a / self.a
+            return approx(other.b, self.b * k) and approx(other.c, self.c * k) and approx(other.d, self.d * k)
+        elif self.b != 0:
+            k = other.b / self.b
+            return approx(other.a, self.a * k) and approx(other.c, self.c * k) and approx(other.d, self.d * k)
+        elif self.c != 0:
+            k = other.c / self.c
+            return approx(other.a, self.a * k) and approx(other.b, self.b * k) and approx(other.d, self.d * k)
+        else:
+            return False
+
     def cal_angle(self, other: 'Line3 | Plane3') -> 'AnyAngle':
         """
         计算平面与平面之间的夹角。
@@ -126,6 +149,16 @@ class Plane3:
         """
         return Plane3.from_point_and_normal(point, self.normal)
 
+    def is_parallel(self, other: 'Plane3') -> bool:
+        """
+        判断两个平面是否平行。
+        Args:
+            other: 另一个平面
+        Returns:
+            是否平行
+        """
+        return self.normal.is_parallel(other.normal)
+
     @property
     def normal(self) -> 'Vector3':
         """
@@ -237,5 +270,10 @@ class Plane3:
         else:
             raise TypeError(f"unsupported operand type(s) for &: 'Plane3' and '{type(other)}'")
 
+    def __eq__(self, other) -> bool:
+        return self.approx(other)
+
     def __rand__(self, other: 'Line3') -> 'Point3':
         return self.cal_intersection_point3(other)
+
+
diff --git a/mbcp/mp_math/point.py b/mbcp/mp_math/point.py
index ff6ce9f..7a1bed6 100644
--- a/mbcp/mp_math/point.py
+++ b/mbcp/mp_math/point.py
@@ -1,5 +1,8 @@
 from typing import TYPE_CHECKING, overload
 
+from .const import APPROX
+from .utils import approx
+
 if TYPE_CHECKING:
     from .vector import Vector3  # type: ignore
 
@@ -17,6 +20,18 @@ class Point3:
         self.y = y
         self.z = z
 
+    def approx(self, other: "Point3", epsilon: float = APPROX) -> bool:
+        """
+        判断两个点是否近似相等。
+        Args:
+            other:
+            epsilon:
+
+        Returns:
+            是否近似相等
+        """
+        return all([abs(self.x - other.x) < epsilon, abs(self.y - other.y) < epsilon, abs(self.z - other.z) < epsilon])
+
     def __str__(self):
         return f"Point3({self.x}, {self.y}, {self.z})"
 
@@ -45,7 +60,7 @@ class Point3:
             other:
         Returns:
         """
-        return self.x == other.x and self.y == other.y and self.z == other.z
+        return approx(self.x, other.x) and approx(self.y, other.y) and approx(self.z, other.z)
 
     def __sub__(self, other: "Point3") -> "Vector3":
         """
diff --git a/mbcp/mp_math/segment.py b/mbcp/mp_math/segment.py
index d4d3afa..4168445 100644
--- a/mbcp/mp_math/segment.py
+++ b/mbcp/mp_math/segment.py
@@ -23,68 +23,18 @@ class Segment3:
         :param p1:
         :param p2:
         """
-        self._start = p1
-        self._end = p2
+        self.p1 = p1
+        self.p2 = p2
 
         """方向向量"""
-        self._direction = self._end - self._start
+        self.direction = self.p2 - self.p1
         """长度"""
-        self._length = self._direction.length
+        self.length = self.direction.length
         """中心点"""
-        self._midpoint = (self._start + self._end) / 2
+        self.midpoint = Point3((self.p1.x + self.p2.x) / 2, (self.p1.y + self.p2.y) / 2, (self.p1.z + self.p2.z) / 2)
+
+    def __repr__(self):
+        return f"Segment3({self.p1}, {self.p2})"
 
     def __str__(self):
-        return f"Segment3({self._start}, {self._end})"
-
-    def _unset_properties(self):
-        self._length = None
-        self._direction = None
-        self._midpoint = None
-
-    @property
-    def start(self) -> "Point3":
-        return self._start
-
-    @start.setter
-    def start(self, value: "Point3"):
-        self._start = value
-        self._unset_properties()
-
-    @property
-    def end(self) -> "Point3":
-        return self._end
-
-    @end.setter
-    def end(self, value: "Point3"):
-        self._end = value
-        self._unset_properties()
-
-    @property
-    def length(self) -> float:
-        """
-        线段的长度。
-        :return:
-        """
-        if self._length is None:
-            self._length = (self._end - self._start).length
-        return self._length
-
-    @property
-    def direction(self) -> "Vector3":
-        """
-        线段的方向向量。
-        :return:
-        """
-        if self._direction is None:
-            self._direction = self._end - self._start
-        return self._direction
-
-    @property
-    def midpoint(self) -> "Point3":
-        """
-        线段的中点。
-        :return:
-        """
-        if self._midpoint is None:
-            self._midpoint = (self._start + self._end) / 2
-        return self._midpoint
+        return f"Segment3({self.p1} -> {self.p2})"
diff --git a/mbcp/mp_math/utils.py b/mbcp/mp_math/utils.py
index ef5c072..ae9f85a 100644
--- a/mbcp/mp_math/utils.py
+++ b/mbcp/mp_math/utils.py
@@ -8,9 +8,16 @@ Copyright (C) 2020-2024 LiteyukiStudio. All Rights Reserved
 @File    : utils.py
 @Software: PyCharm
 """
-from typing import overload
+from typing import overload, TYPE_CHECKING
 
-from mbcp.mp_math.mp_math_typing import RealNumber
+from .mp_math_typing import RealNumber
+from .const import APPROX
+
+if TYPE_CHECKING:
+    from .vector import Vector3
+    from .point import Point3
+    from .plane import Plane3
+    from .line import Line3
 
 
 def clamp(x: float, min_: float, max_: float) -> float:
@@ -27,24 +34,36 @@ def clamp(x: float, min_: float, max_: float) -> float:
     return max(min(x, max_), min_)
 
 
-class Approx(float):
+class Approx:
     """
-    用于近似比较浮点数的类。
-    """
-    epsilon = 0.001
-    """全局近似值。"""
+    用于近似比较对象
 
-    def __new__(cls, x: RealNumber):
-        return super().__new__(cls, x)
+    已实现对象 实数 Vector3 Point3 Plane3 Line3
+    """
+
+    def __init__(self, value: RealNumber):
+        self.value = value
 
     def __eq__(self, other):
-        return abs(self - other) < Approx.epsilon
+        if isinstance(self.value, (float, int)):
+            if isinstance(other, (float, int)):
+                return abs(self.value - other) < APPROX
+            else:
+                self.raise_type_error(other)
+        elif isinstance(self.value, Vector3):
+            if isinstance(other, (Vector3, Point3, Plane3, Line3)):
+                return all([approx(self.value.x, other.x), approx(self.value.y, other.y), approx(self.value.z, other.z)])
+            else:
+                self.raise_type_error(other)
+
+    def raise_type_error(self, other):
+        raise TypeError(f"Unsupported type: {type(self.value)} and {type(other)}")
 
     def __ne__(self, other):
         return not self.__eq__(other)
 
 
-def approx(x: float, y: float = 0.0, epsilon: float = 0.0001) -> bool:
+def approx(x: float, y: float = 0.0, epsilon: float = APPROX) -> bool:
     """
     判断两个数是否近似相等。或包装一个实数，用于判断是否近似于0。
     Args:
diff --git a/mbcp/mp_math/vector.py b/mbcp/mp_math/vector.py
index 6988472..7e1c168 100644
--- a/mbcp/mp_math/vector.py
+++ b/mbcp/mp_math/vector.py
@@ -1,8 +1,12 @@
 import math
 from typing import overload, TYPE_CHECKING
 
+import numpy as np
+
+from .const import APPROX
 from .mp_math_typing import RealNumber
 from .point import Point3
+from .utils import approx
 
 if TYPE_CHECKING:
     from .angle import AnyAngle
@@ -21,6 +25,18 @@ class Vector3:
         self.y = y
         self.z = z
 
+    def approx(self, other: 'Vector3', epsilon: float = APPROX) -> bool:
+        """
+        判断两个向量是否近似相等。
+        Args:
+            other:
+            epsilon:
+
+        Returns:
+            是否近似相等
+        """
+        return all([abs(self.x - other.x) < epsilon, abs(self.y - other.y) < epsilon, abs(self.z - other.z) < epsilon])
+
     def cal_angle(self, other: 'Vector3') -> 'AnyAngle':
         """
         计算两个向量之间的夹角。
@@ -31,16 +47,6 @@ class Vector3:
         """
         return AnyAngle(math.acos(self @ other / (self.length * other.length)), is_radian=True)
 
-    def is_parallel(self, other: 'Vector3') -> bool:
-        """
-        判断两个向量是否平行。
-        Args:
-            other: 另一个向量
-        Returns:
-            是否平行
-        """
-        return self.cross(other) == Vector3(0, 0, 0)
-
     def cross(self, other: 'Vector3') -> 'Vector3':
         """
         向量积 叉乘：v1 cross v2 -> v3
@@ -61,6 +67,27 @@ class Vector3:
                        self.z * other.x - self.x * other.z,
                        self.x * other.y - self.y * other.x)
 
+    def is_approx_parallel(self, other: 'Vector3', epsilon: float = APPROX) -> bool:
+        """
+        判断两个向量是否近似平行。
+        Args:
+            other: 另一个向量
+            epsilon: 允许的误差
+        Returns:
+            是否近似平行
+        """
+        return self.cross(other).length < epsilon
+
+    def is_parallel(self, other: 'Vector3') -> bool:
+        """
+        判断两个向量是否平行。
+        Args:
+            other: 另一个向量
+        Returns:
+            是否平行
+        """
+        return self.cross(other).approx(zero_vector3)
+
     def normalize(self):
         """
         将向量归一化。
@@ -72,6 +99,15 @@ class Vector3:
         self.y /= length
         self.z /= length
 
+    @property
+    def np_array(self) -> 'np.ndarray':
+        """
+        返回numpy数组
+        Returns:
+        """
+
+        return np.array([self.x, self.y, self.z])
+
     @property
     def length(self) -> float:
         """
@@ -90,6 +126,9 @@ class Vector3:
         """
         return self / self.length
 
+    def __abs__(self):
+        return self.length
+
     @overload
     def __add__(self, other: 'Vector3') -> 'Vector3':
         ...
@@ -122,7 +161,7 @@ class Vector3:
         Returns:
             是否相等
         """
-        return self.x == other.x and self.y == other.y and self.z == other.z
+        return approx(self.x, other.x) and approx(self.y, other.y) and approx(self.z, other.z)
 
     def __radd__(self, other: 'Point3') -> 'Point3':
         """
@@ -197,7 +236,7 @@ class Vector3:
     def __rmul__(self, other: 'RealNumber') -> 'Vector3':
         return self.__mul__(other)
 
-    def __matmul__(self, other: 'Vector3') -> float:
+    def __matmul__(self, other: 'Vector3') -> 'RealNumber':
         """
         点乘。
         Args:
diff --git a/tests/answer.py b/tests/answer.py
index 4d67b07..0c2f0da 100644
--- a/tests/answer.py
+++ b/tests/answer.py
@@ -8,16 +8,19 @@ Copyright (C) 2020-2024 LiteyukiStudio. All Rights Reserved
 @File    : .answer.py
 @Software: PyCharm
 """
+from typing import Optional
+
 from liteyuki.log import logger  # type: ignore
 
 
-def output_answer(correct_ans, actual_ans, question: str = None):
+def output_ans(correct_ans, actual_ans, condition: Optional[bool] = None,question: Optional[str] = None):
     """
     输出答案
     Args:
         correct_ans:
         actual_ans:
-        question:
+        condition: 判对条件
+        question: 问题
 
     Returns:
 
@@ -25,8 +28,31 @@ def output_answer(correct_ans, actual_ans, question: str = None):
     print("")
     if question is not None:
         logger.info(f"问题：{question}")
-    r = correct_ans == actual_ans
+    r = (correct_ans == actual_ans) if condition is None else condition
     if r:
         logger.success(f"测试正确    正确答案：{correct_ans}    实际答案：{actual_ans}")
     else:
         logger.error(f"测试错误    正确答案：{correct_ans}    实际答案：{actual_ans}")
+        assert r
+
+
+def output_step_ans(correct_ans, actual_ans, condition: Optional[bool] = None, question: Optional[str] = None):
+    """
+    输出步骤答案
+    Args:
+        correct_ans: 正确答案
+        actual_ans: 实际答案
+        condition: 判对条件
+        question: 问题
+
+    Returns:
+
+    """
+    print("")
+    if question is not None:
+        logger.info(f"  步骤：{question}")
+    r = (correct_ans == actual_ans) if condition is None else condition
+    if r:
+        logger.success(f"  正确    正确：{correct_ans}    实际：{actual_ans}")
+    else:
+        logger.error(f"  错误    正确：{correct_ans}    实际：{actual_ans}")
diff --git a/tests/test_line3.py b/tests/test_line3.py
index 7388e5f..26516c0 100644
--- a/tests/test_line3.py
+++ b/tests/test_line3.py
@@ -13,8 +13,50 @@ import logging
 from mbcp.mp_math.point import Point3
 from mbcp.mp_math.vector import Vector3
 from mbcp.mp_math.line import Line3
-from tests.answer import output_answer
+from tests.answer import output_ans
 
 
+class TestLine3:
 
+    def test_equal(self):
+        line1 = Line3(Point3(1, 1, 1), Vector3(1, 1, 1))
+        line2 = Line3(Point3(1, 1, 1), Vector3(2, 2, 2))
+        output_ans(True, line1 == line2, question="判断两条直线是否相等")
 
+        # 反例
+        line1 = Line3(Point3(1, 1, 1), Vector3(1, 1, 1))
+        line2 = Line3(Point3(1, 1, 1), Vector3(2, 2, 2.000000001))
+        output_ans(False, line1 == line2, question="判断两条直线是否不相等")
+
+    def test_approx(self):
+        line1 = Line3(Point3(1, 1, 1), Vector3(1, 1, 1))
+        line2 = Line3(Point3(1, 1, 1), Vector3(2, 2, 2.000000001))
+        output_ans(True, line1.approx(line2), question="判断两条直线是否近似相等")
+
+        # 反例
+        line1 = Line3(Point3(1, 1, 1), Vector3(1, 1, 1))
+        line2 = Line3(Point3(1, 1, 1), Vector3(2, 2, 3.1))
+        output_ans(False, line1.approx(line2), question="判断两条直线是否不近似相等")
+
+    def test_cal_intersection(self):
+        line1 = Line3.from_two_points(Point3(0, 0, 0), Point3(2, 2, 2))
+        line2 = Line3.from_two_points(Point3(0, 0, 2), Point3(2, 2, 0))
+        output_ans(Point3(1, 1, 1), line1 & line2, question="计算两条直线的交点测1")
+
+        line1 = Line3.from_two_points(Point3(0, 0, 0), Point3(0, 2, 2))
+        line2 = Line3.from_two_points(Point3(0, 0, 2), Point3(0, 2, 0))
+        output_ans(Point3(0, 1, 1), line1 & line2, question="计算两条直线的交点测2")
+
+        line1 = Line3.from_two_points(Point3(0, 0, 0), Point3(0, 0, 2))
+        line2 = Line3.from_two_points(Point3(0, 0, 2), Point3(0, 2, 0))
+        output_ans(Point3(0, 0, 2), line1 & line2, question="计算两条直线的交点测3")
+
+        # 反例：平行线无交点
+        line1 = Line3(Point3(1, 1, 1), Vector3(1, 1, 1))
+        line2 = Line3(Point3(2, 3, 1), Vector3(1, 1, 1))
+        output_ans(None, line1 & line2, question="平行线交集为空集")
+
+        # 反例：重合线交集为自身
+        line1 = Line3(Point3(1, 1, 1), Vector3(1, 1, 1))
+        line2 = Line3(Point3(0, 0, 0), Vector3(2, 2, 2))
+        output_ans(line1, line1 & line2, question="重合线的交集为自身")
diff --git a/tests/test_vector3.py b/tests/test_vector3.py
index 66acead..a4abd6e 100644
--- a/tests/test_vector3.py
+++ b/tests/test_vector3.py
@@ -12,7 +12,7 @@ import logging
 
 from mbcp.mp_math.vector import Vector3
 
-from tests.answer import output_answer
+from tests.answer import output_ans
 
 
 class TestVector3:
@@ -41,7 +41,7 @@ class TestVector3:
         v1 = Vector3(1, 2, 3)
         v2 = Vector3(3, 6, 9)
         actual_ans = v1.is_parallel(v2)
-        output_answer(correct_ans, actual_ans)
+        output_ans(correct_ans, actual_ans)
         assert correct_ans == actual_ans
 
         """小题2"""
@@ -49,7 +49,7 @@ class TestVector3:
         v1 = Vector3(1, 2, 3)
         v2 = Vector3(3, 6, 8)
         actual_ans = v1.is_parallel(v2)
-        output_answer(correct_ans, actual_ans)
+        output_ans(correct_ans, actual_ans)
         assert correct_ans == actual_ans
 
 
diff --git a/tests/test_word_problem.py b/tests/test_word_problem.py
index 56d5ad2..c86dc7e 100644
--- a/tests/test_word_problem.py
+++ b/tests/test_word_problem.py
@@ -7,7 +7,7 @@ from mbcp.mp_math.line import Line3
 from mbcp.mp_math.plane import Plane3
 from mbcp.mp_math.point import Point3
 from mbcp.mp_math.vector import Vector3
-from .answer import output_answer
+from .answer import output_ans, output_step_ans
 
 
 class TestWordProblem:
@@ -27,7 +27,7 @@ class TestWordProblem:
         s = pl1.normal.cross(pl2.normal)
         actual_ans = Line3(p, s)
 
-        output_answer(correct_ans, actual_ans, question)
+        output_ans(correct_ans, actual_ans, question=question)
         assert actual_ans == correct_ans
 
         """解法2"""
@@ -38,7 +38,7 @@ class TestWordProblem:
         # 求pl3和pl4的交线
         actual_ans = pl3.cal_intersection_line3(pl4)
 
-        output_answer(correct_ans, actual_ans, question)
+        output_ans(correct_ans, actual_ans, question=question)
         assert actual_ans == correct_ans
 
     def test_c8s4e5(self):
@@ -56,21 +56,37 @@ class TestWordProblem:
 
         """解"""
         actual_ans = plane & line
-        output_answer(correct_ans, actual_ans, question)
+        output_ans(correct_ans, actual_ans, question=question)
 
     def test_c8s4e6(self):
         question = "求过点(2, 3, 1)且与直线(x+1)/3 = (y-1)/2 = z/-1垂直相交的直线的方程。"
         """正确答案"""
         correct_ans = Line3(Point3(2, 1, 3), Vector3(2, -1, 4))
         """题目已知量"""
-        point = Point3(2, 3, 1)
+        point = Point3(2, 1, 3)
         line = Line3(Point3(-1, 1, 0), Vector3(3, 2, -1))
 
         """解"""
-        # 先作平面过点且垂直与已知直线
-        pl = line.cal_perpendicular(point)
-        logger.debug(line.get_point(1))
+        # 先作过点且垂直与已知直线的平面
+        s1_correct_ans = Plane3(3, 2, -1, -5)
+        pl = Plane3.from_point_and_normal(point, line.direction)
 
-        # output_answer(correct_ans, actual_ans, question)
+        output_step_ans(s1_correct_ans, pl, question="作过点且垂直与已知直线的平面")
 
+        # 求该平面与已知直线的交点
+        s2_correct_ans = Point3(2 / 7, 13 / 7, -3 / 7)
+        s2_actual_ans = pl & line
 
+        output_step_ans(s2_correct_ans, s2_actual_ans, s1_correct_ans.approx(s1_correct_ans), question="求该平面与已知直线的交点")
+
+        # 求所求直线的方向向量
+        s3_correct_ans = (-6 / 7) * Vector3(2, -1, 4)
+
+        dv = s2_correct_ans - point
+
+        output_step_ans(s3_correct_ans, dv, condition=s3_correct_ans.unit.approx(dv.unit), question="求所求直线的方向向量")
+
+        # 求所求直线的方程
+        actual_ans = Line3(point, dv)
+
+        output_ans(correct_ans, actual_ans, correct_ans.approx(actual_ans), question=question)