Coordinates of Points A and B
The coordinates \(x$$ and \(y$$ of points A and B represent the x- and y-axis coordinates, respectively:
$$A(x_1, y_1)$$ $$B(x_2, y_2)$$
Coordinates of Vector AB
$$\overrightarrow{AB}=(X;Y)=(x_2-x_1;y_2-y_1)$$
Length of Vector AB
$$\left |\overrightarrow{AB} \right |=\sqrt{X^{2}+Y^{2}}=\sqrt{\left (x_2-x_1 \right )^{2}+\left (y_2-y_1 \right )^{2}}$$
Coordinates of Midpoint M of Line Segment AB
$$M\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$
Coordinates of Sum and Difference of Two Vectors
$$\vec{a} \pm \vec{b} = (X_1 \pm X_2, Y_1 \pm Y_2)$$
Scalar Multiplication of a Vector
$$c\vec{a}=(cX;cY)$$
Scalar Product of Two Vectors
$$\vec{a}\cdot \vec{b}=X_1X_2+Y_1Y_2$$
Cosine of the Angle Between Two Vectors
$$\cos(\alpha) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \cdot |\vec{b}|}=\frac{X_1 \cdot X_2 + Y_1 \cdot Y_2}{\sqrt{X_1^2 + Y_1^2} \cdot \sqrt{X_2^2 + Y_2^2}}$$
Condition for Two Vectors to be Orthogonal
$$\vec{a} \times \vec{b} = X_1 \cdot Y_2 - Y_1 \cdot X_2 = 0$$
Condition for Two Vectors to be Parallel
$$\frac{X_1}{X_2} = -\frac{Y_1}{Y_2}$$
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