How to find the dot product and length of a vector

A dot product takes two vectors and gives us a single number. We find it by multiplying the components in matching positions and adding the products. Let’s use that calculation to find a dot product, then the length of a vector.

Multiply matching components and add

For two vectors with the same number of components, use $u \cdot v = u_{1} v_{1} + u_{2} v_{2} + \dots + u_{n} v_{n}$ . For example, let $u = (3, 4)$ and $v = (- 1, 2)$ .

\begin{array}{l} u \cdot v = 3 (- 1) + 4 (2) \\ = - 3 + 8 = 5 \end{array}

Dot a vector with itself to find its squared length

If both vectors are $u$ , each product becomes a square. That gives $u \cdot u = {‖ u ‖}^{2}$ . Taking the nonnegative square root gives the length.

\begin{array}{l} (3, 4) \cdot (3, 4) = 3^{2} + 4^{2} = 25 \\ ‖ (3, 4) ‖ = \sqrt{25} = 5 \end{array}

Use the same process in more dimensions

For $w = (2, - 3, 4, 0)$ , square and add all four components.

\begin{array}{l} w \cdot w = 2^{2} + {(- 3)}^{2} \\ + 4^{2} + 0^{2} = 29 \\ ‖ w ‖ = \sqrt{29} \end{array}

The order of a dot product can be reversed, so $u \cdot v = v \cdot u$ . It also distributes across sums of vectors.

Example: Find dot products in two and three dimensions

For $a = (2, 1)$ and $b = (6, - 2)$ , multiply the two pairs of matching components.

a \cdot b = 2 (6) + 1 (- 2) = 10

The same process works with a third component. If $c = (2, - 6, 1)$ and $d = (0, 2, - 3)$ , we get

\begin{array}{l} c \cdot d = 2 (0) + (- 6) (2) \\ + 1 (- 3) = - 15 \end{array}

Use a zero dot product to recognize perpendicular vectors

Two nonzero vectors are perpendicular exactly when their dot product is zero. For example, $(2, 1) \cdot (1, - 2) = 2 - 2 = 0$ , so those directions form a right angle. The zero vector also has dot product zero with every vector, but it has no direction and does not define an angle.

Learn mathKrista KingAugust 18, 2020math, learn online, online course, online math, dot product, calculus 3, calculus iii, calc 3, calc iii, dot product of two vectors, vectors