# The Dot Product

The dot product of two vectors is the product of the two x-components plus the product of the two y-components.

DOT PRODUCT OF VECTORS:

Let$u=〈{u}_{1},{u}_{2}〉$ and $v=〈{v}_{1},{v}_{2}〉$

$u·v={u}_{1}{v}_{1}+{u}_{2}{v}_{2}$

The vector operations of scalar multiplications and vector addition both result in a vector. However, the dot product results in a scalar or a number, not a vector.

The following are properties of the dot product.

PROPERTIES OF THE DOT PRODUCT:

Let u,v and w be vectors and c be scalars.

1. u · v = v · u      2. 0 · v = 0

3. u · (v · w) = u · v + u · w4. v · v = ||v||2

5. c(u · v) = cu · v = u · cv

Let's look at some examples.

To work these examples requires the use of various vector rules. If you are not familiar with a rule go to the associated topic for a review.

Example 1:     Let $u=〈5,7〉$, $v=〈3,1〉$ and $w=〈-6,-4〉$. Find each dot product:
(a) u · v     (b) u · w     (c) v · w
 Step 1: Evaluate u · v. Remember the result will be a scalar. $u·v={u}_{1}{v}_{1}+{u}_{2}{v}_{2}$ $\left(5·3\right)+\left(7·1\right)$ $15+7={22}$ Step 2: Evaluate u · w. Remember the result will be a scalar. $u·w={u}_{1}{w}_{1}+{u}_{2}{w}_{2}$ $\left(5·-6\right)+\left(7·-4\right)$ $-30-28={-}{58}$ Step 3: Evaluate v · w. Remember the result will be a scalar. $v·w={v}_{1}{w}_{1}+{v}_{2}{w}_{2}$ $\left(3·-6\right)+\left(1·-4\right)$ $-18-4={-}{22}$
Example 2:     Let $u=〈2,1〉$, $v=〈10,-3〉$ and $w=〈4,5〉$. Find each dot product:
(a) (v · w) · u     (b) 2v · w     (c) (u + v) · w
 Step 1: Evaluate (v · w) · u using Property 3. Remember the dot product will be a scalar. $\left({u}_{1}{v}_{1}+{u}_{2}{v}_{2}\right)+\left({u}_{1}{w}_{1}+{u}_{2}{w}_{2}\right)$ $\left[\left(2·10\right)+\left(1·-3\right)\right]+\left[\left(2·4\right)+\left(1·5\right)\right]$ $17+13={30}$ Step 2: Evaluate 2v · w. First multiply v by 2 using scalar multiplication and then find the dot product, which will be a scalar result. $〈20,-6〉·〈4,5〉$ $\left(20·4\right)+\left(-6·5\right)$ $80-30={50}$ Step 3: Evaluate (u + v) · w. First add vectors u and v using scalar multiplication then find the dot product of the resultant vector and w. Remember the dot product will be a scalar. $\left(u+v\right)·w$ $\left[〈{u}_{1}+{v}_{1}〉,〈{u}_{2}+{v}_{2}〉\right]·w$ $\left[\left(2+10\right),\left(1+\left(-3\right)\right)\right]·w$ $\left(12,-2\right)·w$ $\left(u+v\right)·w={x}_{1}{w}_{1}+{y}_{2}{w}_{2}$ $\left(12·4\right)+\left(-2·5\right)$ $48+\left(-10\right)={38}$

 Related Links: Math algebra Angle between Two Vectors Decomposing a Vector into Components Pre Calculus

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