Vectors and Matrices Reference

Vectors and Dot Products

A vector records both direction and magnitude. Components make vector operations precise and easy to compute.

Solved Examples

Jump directly to a worked example by vector operation.

Fact Table

TaskRuleWatch For
Components from points$\overrightarrow{PQ}=\langle x_2-x_1,y_2-y_1\rangle$Subtract initial point from terminal point.
Magnitude$\|\langle a,b\rangle\|=\sqrt{a^2+b^2}$Magnitude is nonnegative.
Add vectorsAdd matching components.Do not add magnitudes unless vectors are collinear in the same direction.
Dot product$\langle a,b\rangle\cdot\langle c,d\rangle=ac+bd$A dot product is a number, not a vector.

Content Formulas

Scalar Multiple
$$k\langle a,b\rangle=\langle ka,kb\rangle$$
Dot Product
$$\vec u\cdot\vec v=\|\vec u\|\|\vec v\|\cos\theta$$
Perpendicular Test
$$\vec u\cdot\vec v=0\Longrightarrow\vec u\perp\vec v$$

Classic Examples

Find Components and Magnitude

Find the component form and magnitude of the vector from $P(1,2)$ to $Q(4,6)$.

Solution Steps
  1. Subtract the coordinates of $P$ from the coordinates of $Q$.
  2. Use the component values in the magnitude formula.
  3. Simplify the square root.
$$\overrightarrow{PQ}=\langle4-1,6-2\rangle=\langle3,4\rangle$$$$\|\overrightarrow{PQ}\|=\sqrt{3^2+4^2}$$$$\boxed{\langle3,4\rangle,\quad\|\overrightarrow{PQ}\|=5}$$

Add Vectors

Find $\langle2,-1\rangle+\langle-3,4\rangle$.

Solution Steps
  1. Add the horizontal components.
  2. Add the vertical components.
  3. Write the result as one vector.
$$2+(-3)=-1$$$$-1+4=3$$$$\boxed{\langle-1,3\rangle}$$

Find a Dot Product

Find $\langle2,1\rangle\cdot\langle3,-2\rangle$.

Solution Steps
  1. Multiply matching components.
  2. Add the products.
  3. State the scalar result.
$$2(3)+1(-2)$$$$6-2=4$$$$\boxed{4}$$

Test Perpendicular Vectors

Are $\langle3,4\rangle$ and $\langle4,-3\rangle$ perpendicular?

Solution Steps
  1. Find the dot product.
  2. Check whether it is zero.
  3. State the geometric conclusion.
$$3(4)+4(-3)=12-12$$$$\langle3,4\rangle\cdot\langle4,-3\rangle=0$$$$\boxed{\text{The vectors are perpendicular.}}$$