# Vector in the Plane

## 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

$$\frac{X_1}{X_2} = -\frac{Y_1}{Y_2}$$