Skill: Determine the dot product of two vectors

To find the dot product, multiply corresponding components and add the results.

For two-dimensional vectors,

$$\mathbf{u}\cdot\mathbf{v}=u_xv_x+u_yv_y.$$

For three-dimensional vectors, include the third pair:

$$\mathbf{u}\cdot\mathbf{v}=u_xv_x+u_yv_y+u_zv_z.$$

Example

Find the dot product of

$$\mathbf{u}=(3,-2) \qquad\text{and}\qquad \mathbf{v}=(4,5).$$

Step 1: Multiply corresponding components.

$$3(4)=12 \qquad\text{and}\qquad (-2)(5)=-10$$

Step 2: Add the products.

$$\mathbf{u}\cdot\mathbf{v}=12+(-10)=2$$

Therefore,

$$\boxed{\mathbf{u}\cdot\mathbf{v}=2}$$

The dot product is a scalar, not a vector. Its sign also gives information about the angle between the vectors:

• A positive dot product means the angle is acute.
• A dot product of $0$ means the vectors are perpendicular.
• A negative dot product means the angle is obtuse.

Here, the dot product is positive, so the angle between $\mathbf{u}$ and $\mathbf{v}$ is acute.