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 $v = 〈 v_{1}, v_{2} 〉$

$u \cdot v = u_{1} v_{1} + u_{2} v_{2}$

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.

$u \cdot v = u_{1} v_{1} + u_{2} v_{2}$

$(5 \cdot 3) + (7 \cdot 1)$

$15 + 7 = 22$

Step 2: Evaluate u · w.

Remember the result will be a scalar.

$u \cdot w = u_{1} w_{1} + u_{2} w_{2}$

$(5 \cdot - 6) + (7 \cdot - 4)$

$- 30 - 28 = - 58$

Step 3: Evaluate v · w.

Remember the result will be a scalar.

$v \cdot w = v_{1} w_{1} + v_{2} w_{2}$

$(3 \cdot - 6) + (1 \cdot - 4)$

$- 18 - 4 = - 22$

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.

$(v \cdot w) u = u \cdot v + u \cdot w$

$(u_{1} v_{1} + u_{2} v_{2}) + (u_{1} w_{1} + u_{2} w_{2})$

$[(2 \cdot 10) + (1 \cdot - 3)] + [(2 \cdot 4) + (1 \cdot 5)]$

$17 + 13 = 30$

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.

$2 v \cdot w = 2 〈 10, - 3 〉 \cdot 〈 4, 5 〉$

$〈 20, - 6 〉 \cdot 〈 4, 5 〉$

$(20 \cdot 4) + (- 6 \cdot 5)$

$80 - 30 = 50$

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.

$(u + v) \cdot w$

$[〈 u_{1} + v_{1} 〉, 〈 u_{2} + v_{2} 〉] \cdot w$

$[(2 + 10), (1 + (- 3))] \cdot w$

$(12, - 2) \cdot w$

$(u + v) \cdot w = x_{1} w_{1} + y_{2} w_{2}$

$(12 \cdot 4) + (- 2 \cdot 5)$

$48 + (- 10) = 38$

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

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