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 [latex]\overrightarrow{u}=\lt x_{1},y_{1}\gt[/latex] and [latex]\overrightarrow{v}=\lt x_{2},y_{2}\gt[/latex] then the dot product of [latex]\overrightarrow{u}[/latex] and [latex]\overrightarrow{v}[/latex] is a number given by [latex]\overrightarrow{u}\cdot\overrightarrow{v}=\lt x_{1},y_{1}\gt\cdot\lt x_{2},y_{2}\gt=x_{1}x_{2}+y_{1}y_{2}[/latex].
(b) If [latex]\overrightarrow{u}=\lt x_{1},y_{1},z_{1}\gt[/latex] and [latex]\overrightarrow{v}=\lt x_{2},y_{2},z_{2}\gt[/latex] then the dot product of [latex]\overrightarrow{u}[/latex] and [latex]\overrightarrow{v}[/latex] is a number given by [latex]\overrightarrow{u}\cdot\overrightarrow{v}=\lt x_{1},y_{1},z_{1}\gt\cdot\lt x_{2},y_{2},z_{2}\gt=x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}[/latex].
(c) The dot product is often called the scalar product. It may also be called the inner product.
EQ1: The dot product of [latex]\overrightarrow{u}=\lt 1,-2,3\gt[/latex] and [latex]\overrightarrow{v}=\lt 2,1,0\gt[/latex] is
A: [latex]2-2+0=0[/latex]; B: [latex]\lt 2,-2,0\gt[/latex]; C: [latex]1+4-3=2[/latex]; D: [latex]\lt 1,3,3\gt[/latex]
EQ2: The scalar product of [latex]\overrightarrow{u}=3\overrightarrow{i}+4\overrightarrow{j}-5\overrightarrow{k}[/latex] and [latex]\overrightarrow{v}=-\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k}[/latex] is
A: [latex]-3\overrightarrow{i}+8\overrightarrow{j}+5\overrightarrow{k}[/latex]; B: [latex]-3\overrightarrow{i}+8\overrightarrow{j}-5\overrightarrow{k}[/latex]; C: [latex]-3+8-5=0[/latex]; D: [latex]-3+8+5=10[/latex]
Theorem: Properties of dot product
Let [latex]\overrightarrow{u},\overrightarrow{v},[/latex] and [latex]\overrightarrow{w}[/latex] be vectors. Let [latex]r[/latex] be scalars.
i. [latex]\overrightarrow{u}\cdot\overrightarrow{v}=\overrightarrow{v}\cdot\overrightarrow{u}[/latex] Commutative property
ii . [latex]\overrightarrow{u}\cdot(\overrightarrow{v}+\overrightarrow{w})=\overrightarrow{u}\cdot\overrightarrow{v}+\overrightarrow{u}\cdot\overrightarrow{w}[/latex] Distributive property
iii. [latex]c(\overrightarrow{u}\cdot\overrightarrow{v}) =(c\overrightarrow{u})\cdot\overrightarrow{v}=\overrightarrow{u}\cdot(c\overrightarrow{v})[/latex] Associative property
iv. [latex]\overrightarrow{v}\cdot\overrightarrow{v}=||\overrightarrow{v}||^{2}[/latex]Property of magnitude.
Proof of iv:
Example 1: Let [latex]\overrightarrow{u}=\lt 1,-2,3\gt,\overrightarrow{v}=\lt 2,1,0\gt[/latex], and [latex]\overrightarrow{w}=\lt -2,0,1\gt[/latex] be vectors.
(a) [latex](\overrightarrow{u}\cdot\overrightarrow{v})\overrightarrow{w}[/latex]
(b) [latex]3\overrightarrow{u}\cdot\overrightarrow{v}[/latex]
(c) [latex]||\overrightarrow{u}||[/latex]
(d) [latex](\overrightarrow{u}+\overrightarrow{v})\cdot\overrightarrow{w}[/latex]
Exercise 1: Let [latex]\overrightarrow{u}=\lt -1,2,-3\gt,\overrightarrow{v}=\lt -2,0,1\gt[/latex], and [latex]\overrightarrow{w}=\lt 3,2,-1\gt[/latex] be vectors.
(a) [latex](\overrightarrow{u}\cdot\overrightarrow{v})\overrightarrow{w}[/latex]
(b) [latex]3\overrightarrow{u}\cdot\overrightarrow{v}[/latex]
(c) [latex]||\overrightarrow{u}||[/latex]
(d) [latex](\overrightarrow{u}+\overrightarrow{v})\cdot\overrightarrow{w}[/latex]
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 [latex]||\overrightarrow{u}-\overrightarrow{v}||^{2}[/latex]
Example 2: Find the measure of the angle between each pair of vectors.
a. [latex]\overrightarrow{i}+\overrightarrow{j}[/latex] and [latex]\overrightarrow{j}-\overrightarrow{k}[/latex]
b. [latex]\lt 1,2,-3\gt[/latex] and [latex]\lt 3,0,1\gt[/latex]
Exercise 2: Find the measure of the angle between each pair of vectors.
a. [latex]\overrightarrow{i}-\overrightarrow{j}+2\overrightarrow{k}[/latex] and [latex]\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k}[/latex]
b. [latex]\lt -1,2,-3\gt[/latex] and [latex]\lt 0,3,2\gt[/latex]
Theorem: Orthogonal Vectors
The nonzero vectors [latex]\overrightarrow{u}[/latex] and [latex]\overrightarrow{v}[/latex] are orthogonal vectors if and only if [latex]\overrightarrow{u}\cdot\overrightarrow{v}=0.[/latex]
EQ3: Which of the following pair of vectors are NOT orthogonal to each others.
A: [latex]\overrightarrow{j}=\lt 0,1,0\gt[/latex] and [latex]\overrightarrow{i}=\lt 1,0,0\gt[/latex];
B: [latex]\overrightarrow{u}=\lt 0,-1,1\gt[/latex] and [latex]\overrightarrow{v}=\lt 1,0,0\gt[/latex];
C: [latex]\overrightarrow{u}=\lt 1,-1,1\gt[/latex] and [latex]\overrightarrow{v}=\lt 1,1,1\gt[/latex];
D: [latex]\overrightarrow{u}=\lt 1,0,1\gt[/latex] and [latex]\overrightarrow{v}=\lt 0,1,0\gt[/latex]
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 [latex]\overrightarrow{v}[/latex] onto [latex]\overrightarrow{u}[/latex] is the vector labeled [latex]\text{proj}_{\overrightarrow{u}}\overrightarrow{v}[/latex].
It has the same initial point as [latex]\overrightarrow{u}[/latex] and [latex]\overrightarrow{v}[/latex] and it represents the component of [latex]\overrightarrow{v}[/latex] that acts in the direction of [latex]\overrightarrow{u}[/latex]. If [latex]\theta[/latex] represents the angle between [latex]\overrightarrow{u}[/latex] and [latex]\overrightarrow{v}[/latex], then the scalar projection of [latex]\overrightarrow{v}[/latex] onto [latex]\overrightarrow{u}[/latex] is [latex]\text{comp}_{\overrightarrow{u}}\overrightarrow{v}=||\overrightarrow{v}||\text{cos}(\theta)[/latex].
We use dot product to replace [latex]\text{cos}(\theta)[/latex], hence [latex]\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}||}[/latex].
\[\text{proj}_{\overrightarrow{u}}\overrightarrow{v} =(\text{comp}_{\overrightarrow{u}}\overrightarrow{v})(\text{unit vector of }\overrightarrow{u})\]
\[=\frac{\overrightarrow{u}\cdot\overrightarrow{v}}{||\overrightarrow{u}||}\frac{\overrightarrow{u}}{||\overrightarrow{u}||}\]
\[=\frac{(\overrightarrow{u}\cdot\overrightarrow{v})\overrightarrow{u}}{||\overrightarrow{u}||^{2}}\]
\[=\frac{(\overrightarrow{u}\cdot\overrightarrow{v})\overrightarrow{u}}{\overrightarrow{u}\cdot\overrightarrow{u}}\]
Example 3: The vector projection of [latex]\overrightarrow{v}[/latex] onto [latex]\overrightarrow{u}[/latex]
a. [latex]\overrightarrow{u}=\overrightarrow{i}+\overrightarrow{j}[/latex] and [latex]\overrightarrow{v}=\overrightarrow{j}-\overrightarrow{k}[/latex]
b. [latex]\overrightarrow{u}=\lt 1,2,-3\gt[/latex] and [latex]\overrightarrow{v}=\lt 3,0,1\gt[/latex]
Exercise 3: The vector projection of [latex]\overrightarrow{v}[/latex] onto [latex]\overrightarrow{u}[/latex]
a. [latex]\overrightarrow{u}=\overrightarrow{i}-\overrightarrow{j}+2\overrightarrow{k}[/latex] and [latex]\overrightarrow{v}=\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k}[/latex]
b. [latex]\overrightarrow{u}=\lt -1,2,-3\gt[/latex] and [latex]\overrightarrow{v}=\lt 0,3,2\gt[/latex]
Definition
When a constant force is applied to an object so the object moves in a straight line from point [latex]A[/latex] to point [latex]B[/latex], the work
[latex]W[/latex] done by the force [latex]\overrightarrow{F}[/latex], acting at an angle [latex]\theta[/latex] from the line of motion, is given by [latex]W=\|\overrightarrow{F}\|\text{cos}(\theta)||\overrightarrow{AB}||=\overrightarrow{F}\cdot\overrightarrow{AB}[/latex].
EQ4: A conveyor belt generates a force [latex]\overrightarrow{F}=5\overrightarrow{i}-3\overrightarrow{j}+\overrightarrow{k}[/latex] that moves a suitcase from point [latex](1,1,1)[/latex] to point [latex](4,5,6)[/latex] 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: [latex]15-12+5=8J[/latex]; B: [latex]20-15+6=11J[/latex];C: [latex]5-3+1=3J[/latex];
D: [latex]25-18+7=24J[/latex] .
Example 4: Let [latex]\overrightarrow{u}=\lt 2,0,-1\gt[/latex]. Find a vector [latex]\overrightarrow{v}[/latex] such that [latex]\text{comp}_{\overrightarrow{u}}\overrightarrow{v}=2[/latex].
Example 5: 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?
Group work:
1. Determine the real number [latex]c[/latex] such that vectors [latex]\overrightarrow{u}=\lt 2,3\gt[/latex] and [latex]\overrightarrow{v}=\lt 4,c\gt[/latex] are orthogonal.
2. Let [latex]\overrightarrow{u}=\lt 1,-2,0\gt[/latex]. Find a vector [latex]\overrightarrow{v}[/latex] such that [latex]\text{comp}_{\overrightarrow{u}}\overrightarrow{v}=3[/latex].
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 [latex]100\text{N}[/latex] on a rope that makes an angle of [latex]\frac{\pi}{6}[/latex] with the horizontal. Find the work done in pulling the sled [latex]30\text{m}[/latex].
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?