以同一点 O𝑂 为起点的两个已知向量 →a𝑎→， →b𝑏→，以 OA𝑂𝐴，OB𝑂𝐵 为邻边作 □OACB◻𝑂𝐴𝐶𝐵，则以 O𝑂 为起点的向量 −−→OC𝑂𝐶→
（OC𝑂𝐶 是 □OACB◻𝑂𝐴𝐶𝐵 的对角线）就是向量 →a𝑎→ 与 →b𝑏→ 的和.

1.正方形加法

from manim import *

class ParallelogramRule003(Scene):
    def construct(self):
        numberplane = NumberPlane()
        dot = Dot(ORIGIN)
        self.add(numberplane,dot)
        # Create vectors
        vec_a = Arrow(start=dot, end=[1, 3, 0],buff=0,  color=BLUE)
        self.play(Create(vec_a))
        self.wait(1)
        t0 = MathTex(r"\begin{bmatrix}1\\3\end{bmatrix}")
        origin_text_a = t0.next_to([1, 3, 0], LEFT)
        self.play(Create(origin_text_a))
        
        
        vec_b = Arrow(start=dot, end=[3, 0, 0],buff=0, color=RED)
        self.play(Create(vec_b))
        self.wait(1)
        t1 = MathTex(r"\begin{bmatrix}3\\0\end{bmatrix}")
        origin_text_b = t1.next_to([3, 0, 0], DOWN)
        self.play(Create(origin_text_b))
        
        # 创建虚线箭头
        dashed_arrow_a = DashedLine(start=ORIGIN, end=[1, 3, 0], dash_length=0.05, color=BLUE)
        self.add(dashed_arrow_a)
        # 平移虚线箭头到目标位置
        self.play(ApplyMethod(dashed_arrow_a.shift, RIGHT * 3))
        self.wait(1)
        
        dashed_arrow_b = DashedLine(start=ORIGIN, end=[3, 0, 0], dash_length=0.05, color=BLUE)
        self.add(dashed_arrow_b)
        self.play(ApplyMethod(dashed_arrow_b.shift, RIGHT+UP * 3))
        self.wait(1)

    

        
        vec_ac = Arrow(start=dot, end=[4, 3, 0],buff=0,  color=GOLD)
        self.play(Create(vec_ac))
        t2 = MathTex(r"\begin{bmatrix}3 & 4 \\ 0 & 0 \end{bmatrix}")
        origin_text_ac = t2.next_to([4, 3, 0], RIGHT)
        self.play(Create(origin_text_ac))
        
        #self.play(Create(start_arrow))

        self.wait(2)

 实现视频链接如下：

三角形加法 

 已知向量非零向量 →a𝑎→， →b𝑏→，在平面内取任意一点 A𝐴 ， 作 −−→AB=→a𝐴𝐵→=𝑎→ ， −−→BC=→b𝐵𝐶→=𝑏→，则向量 −−→AC𝐴𝐶→ 叫做 →a𝑎→ 与 →b𝑏→ 的和，记作 →a+→b𝑎→+𝑏→，即 →a+→b=−−→AB+−−→BC=−−→AC𝑎→+𝑏→=𝐴𝐵→+𝐵𝐶→=𝐴𝐶→.

from manim import *

class ParallelogramRule004(Scene):
    def construct(self):
        numberplane = NumberPlane()
        dot = Dot(ORIGIN)
        self.add(numberplane,dot)
        # Create vectors
        vec_a = Arrow(start=dot, end=[1, 3, 0],buff=0,  color=BLUE)
        self.play(Create(vec_a))
        self.wait(1)
        t0 = MathTex(r"\begin{bmatrix}1\\3\end{bmatrix}")
        origin_text_a = t0.next_to([1, 3, 0], LEFT)
        self.play(Create(origin_text_a))
        
        
        vec_b = Arrow(start=dot, end=[3, 0, 0],buff=0, color=RED)
        self.play(Create(vec_b))
        self.wait(1)
        t1 = MathTex(r"\begin{bmatrix}3\\0\end{bmatrix}")
        origin_text_b = t1.next_to([3, 0, 0], DOWN)
        self.play(Create(origin_text_b))
        
        # 移动向量a
        # 平移虚线箭头到目标位置
        self.play(ApplyMethod(vec_a.shift, RIGHT * 3))
        self.wait(1)
        
        vec_ac = Arrow(start=dot, end=[4, 3, 0],buff=0,  color=GOLD)
        self.play(Create(vec_ac))
        t2 = MathTex(r"\begin{bmatrix}3 & 4 \\ 0 & 0 \end{bmatrix}")
        origin_text_ac = t2.next_to([4, 3, 0], RIGHT)
        self.play(Create(origin_text_ac))
        
        #self.play(Create(start_arrow))

        self.wait(2)
        

视频实现如下：