Section 1.3 Dot Product

1.3 Dot Product

In this section and next section, we introduce the two most important operations of vectors, dot product and cross product. 

Definition: 

(a) If $\overrightarrow{u}=<x_1,y_1>$ and $\overrightarrow{v}=<x_2,y_2>$ then the dot product of $\overrightarrow{u}$ and $\overrightarrow{v}$  is a number given by $\overrightarrow{u}\cdot \overrightarrow{v}=<x_1,y_1>\cdot <x_2,y_2>=x_1{x}_2+y_1{y}_2$.

(b) If $\overrightarrow{u}=<x_1,y_1,z_1>$ and $\overrightarrow{v}=<x_2,y_2,z_2>$ then the dot product of $\overrightarrow{u}$ and $\overrightarrow{v}$ is a number given by $\overrightarrow{u}\cdot \overrightarrow{v}=<x_1,y_1,z_1>\cdot <x_2,y_2,z_2>=x_1{x}_2+y_1{y}_2+z_1{z}_2$.

(c) The dot product is often called the scalar product. It may also be called the inner product.

EQ1: The dot product of $\overrightarrow{u}=<1,-2,3>$ and $\overrightarrow{v}=<2,1,0>$ is 

A: $2-2+0=0$; B: $<2,-2,0>$; C: $1+4-3=2$; D: $<1,3,3>$

EQ2: The scalar product of $\overrightarrow{u}=3\overrightarrow{i}+4\overrightarrow{j}-5\overrightarrow{k}$ and $\overrightarrow{v}=-\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k}$ is 

A: $-3\overrightarrow{i}+8\overrightarrow{j}+5\overrightarrow{k}$; B: $-3\overrightarrow{i}+8\overrightarrow{j}-5\overrightarrow{k}$; C: $-3+8-5=0$; D: $-3+8+5=10$

Theorem: Properties of dot product 

Let $\overrightarrow{u},\overrightarrow{v},$ and $\overrightarrow{w}$ be vectors. Let $r$ be scalars. 

i. $\overrightarrow{u}\cdot \overrightarrow{v}=\overrightarrow{v}\cdot \overrightarrow{u}$ Commutative property 

ii . $\overrightarrow{u}\cdot (\overrightarrow{v}+\overrightarrow{w})=\overrightarrow{u}\cdot \overrightarrow{v}+\overrightarrow{u}\cdot \overrightarrow{w}$ Distributive property

iii. $c(\overrightarrow{u}\cdot \overrightarrow{v})=(c\overrightarrow{u})\cdot \overrightarrow{v}=\overrightarrow{u}\cdot (c\overrightarrow{v})$ Associative property

iv. $\overrightarrow{v}\cdot \overrightarrow{v}=||\overrightarrow{v}|{|}^2$Property of magnitude.

Proof of iv:

Example 1: Let $\overrightarrow{u}=<1,-2,3>,\overrightarrow{v}=<2,1,0>$, and $\overrightarrow{w}=<-2,0,1>$ be vectors. 

(a) $(\overrightarrow{u}\cdot \overrightarrow{v})\overrightarrow{w}$

(b) $3\overrightarrow{u}\cdot \overrightarrow{v}$

(c) $||\overrightarrow{u}||$

(d) $(\overrightarrow{u}+\overrightarrow{v})\cdot \overrightarrow{w}$

Exercise 1: Let $\overrightarrow{u}=<-1,2,-3>,\overrightarrow{v}=<-2,0,1>$, and $\overrightarrow{w}=<3,2,-1>$ be vectors. 

(a) $(\overrightarrow{u}\cdot \overrightarrow{v})\overrightarrow{w}$

(b) $3\overrightarrow{u}\cdot \overrightarrow{v}$

(c) $||\overrightarrow{u}||$

(d) $(\overrightarrow{u}+\overrightarrow{v})\cdot \overrightarrow{w}$

Theorem:

The dot product of two vectors is the product of the magnitude of each vector and the cosine of the angle between them: 

$$\overrightarrow{u}\cdot \overrightarrow{v}=||\overrightarrow{u}||||\overrightarrow{v}||\text{cos}(\theta ).$$

Proof: Use $||\overrightarrow{u}-\overrightarrow{v}|{|}^2$

Example 2: Find the measure of the angle between each pair of vectors. 

a. $\overrightarrow{i}+\overrightarrow{j}$ and $\overrightarrow{j}-\overrightarrow{k}$

b. $<1,2,-3>$ and $<3,0,1>$

Exercise 2: Find the measure of the angle between each pair of vectors. 

a. $\overrightarrow{i}-\overrightarrow{j}+2\overrightarrow{k}$ and $\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k}$ 

b. $<-1,2,-3>$ and $<0,3,2>$

Theorem: Orthogonal Vectors

The nonzero vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ are orthogonal vectors if and only if $\overrightarrow{u}\cdot \overrightarrow{v}=0.$

EQ3: Which of the following pair of vectors are NOT orthogonal to each others.

A: $\overrightarrow{j}=<0,1,0>$ and $\overrightarrow{i}=<1,0,0>$;

B: $\overrightarrow{u}=<0,-1,1>$ and $\overrightarrow{v}=<1,0,0>$;

C: $\overrightarrow{u}=<1,-1,1>$ and $\overrightarrow{v}=<1,1,1>$;

D: $\overrightarrow{u}=<1,0,1>$ and $\overrightarrow{v}=<0,1,0>$ 

Definition 

The angles formed by a nonzero vector and the coordinate axes are called the direction angles for the vector. The cosines for these angles are called the direction cosines.

Definition 

The vector projection of $\overrightarrow{v}$ onto $\overrightarrow{u}$ is the vector labeled ${\text{proj}}_{\overrightarrow{u}}\overrightarrow{v}$.

It has the same initial point as $\overrightarrow{u}$ and $\overrightarrow{v}$ and it represents the component of $\overrightarrow{v}$ that acts in the direction of $\overrightarrow{u}$. If $\theta$ represents the angle between $\overrightarrow{u}$ and $\overrightarrow{v}$, then the scalar projection of $\overrightarrow{v}$ onto $\overrightarrow{u}$ is ${\text{comp}}_{\overrightarrow{u}}\overrightarrow{v}=||\overrightarrow{v}||\text{cos}(\theta )$.

We use dot product to replace $\text{cos}(\theta )$, hence ${\text{comp}}_{\overrightarrow{u}}\overrightarrow{v}=||\overrightarrow{v}||\frac{\overrightarrow{u}\cdot \overrightarrow{v}}{||\overrightarrow{u}||||\overrightarrow{v}||}=\frac{\overrightarrow{u}\cdot \overrightarrow{v}}{||\overrightarrow{u}||}$.

$${\text{proj}}_{\overrightarrow{u}}\overrightarrow{v}=({\text{comp}}_{\overrightarrow{u}}\overrightarrow{v})(\text{unit vector of }\overrightarrow{u})$$

$$=\frac{\overrightarrow{u}\cdot \overrightarrow{v}}{||\overrightarrow{u}||}\frac{\overrightarrow{u}}{||\overrightarrow{u}||}$$

$$=\frac{(\overrightarrow{u}\cdot \overrightarrow{v})\overrightarrow{u}}{||\overrightarrow{u}|{|}^2}$$

$$=\frac{(\overrightarrow{u}\cdot \overrightarrow{v})\overrightarrow{u}}{\overrightarrow{u}\cdot \overrightarrow{u}}$$

Example 3: The vector projection of $\overrightarrow{v}$ onto $\overrightarrow{u}$ 

a. $\overrightarrow{u}=\overrightarrow{i}+\overrightarrow{j}$ and $\overrightarrow{v}=\overrightarrow{j}-\overrightarrow{k}$ 

b. $\overrightarrow{u}=<1,2,-3>$ and $\overrightarrow{v}=<3,0,1>$

Exercise 3: The vector projection of $\overrightarrow{v}$ onto $\overrightarrow{u}$ 

a. $\overrightarrow{u}=\overrightarrow{i}-\overrightarrow{j}+2\overrightarrow{k}$ and $\overrightarrow{v}=\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k}$ 

b. $\overrightarrow{u}=<-1,2,-3>$ and $\overrightarrow{v}=<0,3,2>$

Definition 

When a constant force is applied to an object so the object moves in a straight line from point $A$ to point $B$, the work

$W$ done by the force $\overrightarrow{F}$, acting at an angle $\theta$ from the line of motion, is given by $W=\parallel \overrightarrow{F}\parallel \text{cos}(\theta )||\overrightarrow{AB}||=\overrightarrow{F}\cdot \overrightarrow{AB}$. 

EQ4: A conveyor belt generates a force $\overrightarrow{F}=5\overrightarrow{i}-3\overrightarrow{j}+\overrightarrow{k}$ that moves a suitcase from point $(1,1,1)$ to point $(4,5,6)$ along a straight line. Find the work done by the conveyor belt. The distance is measured in meters and the force is measured in newtons.

A: $15-12+5=8J$; B: $20-15+6=11J$;C: $5-3+1=3J$; D: $25-18+7=24J$ .

Group work:

1. Determine the real number $c$ such that vectors $\overrightarrow{u}=<2,3>$ and $\overrightarrow{v}=<4,c>$ are orthogonal.

2. Let $\overrightarrow{u}=<1,-2,0>$. Find a vector $\overrightarrow{v}$ such that ${\text{comp}}_{\overrightarrow{u}}\overrightarrow{v}=3$. 

3. A container ship leaves port traveling 15° north of east. Its engine generates a speed of 30 knots along that path. In addition, the ocean current moves the ship southeast at a speed of 3 knots. Considering both the engine and the current, how fast is the ship moving in the direction 15° north of east? 

4. A sled is pulled by exerting a force of $100\text{N}$ on a rope that makes an angle of $\frac{\pi }{6}$ with the horizontal. Find the work done in pulling the sled $30\text{m}$.

5. A boat sails north aided by a wind blowing in a direction of N30°E with a magnitude of 500 lb. How much work is performed by the wind as the boat moves 100 ft? 

6. Let $\overrightarrow{u}=<2,0,-1>$. Find a vector $\overrightarrow{v}$ such that ${\text{comp}}_{\overrightarrow{u}}\overrightarrow{v}=2$.

7. A container ship leaves port traveling 15° north of east. Its engine generates a speed of 20 knots along that path. In addition, the ocean current moves the ship northeast at a speed of 2 knots. Considering both the engine and the current, how fast is the ship moving in the direction 15° north of east?