V3 Scalar product

What is a scalar product? What is a dot product? This is the result of multiplying the magnitudes of the components of two or more vectors. The result is not a vector, but a scalar (which is without direction).

There are two ways to multiply two vectors:

1. The scalar or dot product which gives a number;

2. The vector or cross product which gives a vector.

In this module we consider the scalar or dot product.

Definition

The scalar, or dot, product of two vectors $$\vec{a}\left(a_{1},a_{2},a_{3}\right)$$ and $$\vec{b}\left(b_{1},b_{2},b_{3}\right)$$ is a scalar, defined by: $\vec{a}\cdot\vec{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$ or geometrically, $\vec{a}\cdot\vec{b}=\left|\vec{a}\right|\left|\vec{b}\right|\cos\theta$ where $$\theta$$ is the angle between $$\vec{a}$$ and $$\vec{b}$$.

Properties of the Scalar or Dot Product

1. If $$\vec{a}$$ and $$\vec{b}$$ are non-zero vectors and $$\vec{a}$$ is perpendicular1 Perpendicular means at right angles to. A right angle is $$90^{\circ}=\pi/2.$$ to $$\vec{b}$$ then $$\vec{a}\cdot\vec{b}=0,$$ since $$\cos\left(\frac{\pi}{2}\right)=0$$.

2. If $$\vec{a}$$ is parallel to $$\vec{b}$$ then the angle between the vectors is $$0$$ and $$\vec{a}\cdot\vec{b}=\mid\vec{a}\mid\mid\vec{b}\mid$$ as $$\cos\left(0\right)=1$$.

3. The dot product does not depend on the order of multiplication: $\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}$

4. In three dimensions with $$\hat{i},$$ $$\hat{j}$$ and $$\hat{k}$$ unit vectors along the $$x$$, $$y$$ and $$z$$ axes respectively, we have: \begin{align*} \vec{i}\cdot\vec{j} & =\vec{j}\cdot\vec{k}=\vec{k}\cdot\vec{i}=0\\ \vec{i}\cdot\vec{i} & =\vec{j}\cdot\vec{j}=\vec{k}\cdot\vec{k}=1 \end{align*}

Examples

1. $$\left(2\vec{i}+3\vec{j}+4\vec{k}\right)\cdot\left(-\vec{i}-2\vec{j}+\vec{k}\right)=\left(2\times\left(-1\right)\right)+\left(3\times\left(-2\right)\right)+\left(4\times1\right)=-4$$

2. $$\left(2,-3,-3\right)\cdot\left(1,1,-2\right)=2-3+6=5$$

3. $$\left(5,0,-1\right)\cdot\left(1,4,3\right)=5+0-3=2$$

4. $$\left(2\vec{i}+4\vec{k}\right)\cdot\left(-3\vec{i}-2\vec{j}\right)=2\times\left(-3\right)+0\times\left(-2\right)+4\times0=-6$$

See Exercises 1, 2, and 3.

Angle Between Two Vectors

The angle $$\theta,\left(0\leq\theta\leq\pi\right)$$, between two vectors can be found using the definition of the dot product: \begin{align*} \vec{a}\cdot\vec{b} & =\mid\vec{a}\mid\mid\vec{b}\mid\cos\theta. \end{align*} Rearranging, \begin{align*} \cos\theta & =\frac{\vec{a}\cdot\vec{b}}{\mid\vec{a}\mid\mid\vec{b}\mid} \end{align*} and \begin{align*} \theta & =\cos^{-1}\left(\frac{\vec{a}\cdot\vec{b}}{\mid\vec{a}\mid\mid\vec{b}\mid}\right). \end{align*}

Examples

1. If $$\vec{a}=\left(2,3,1\right)$$ and $$\vec{b}=\left(5,-2,2\right)$$ find the angle $$\theta$$, between $$\vec{a}$$ and $$\vec{b}$$ \begin{align*} \theta & =\cos^{-1}\left(\frac{\vec{a}\cdot\vec{b}}{\mid\vec{a}\mid\mid\vec{b}\mid}\right)\\ \vec{a}\cdot\vec{b} & =\left(2,3,1\right)\cdot\left(5,-2,2\right)=6\\ \mid\vec{a}\mid & =\sqrt{2^{2}+3^{2}+1^{2}}=\sqrt{14},\mid\vec{b}\mid=\sqrt{25+4+4}=\sqrt{33}\\ \theta & =\cos^{-1}\left(\frac{6}{\sqrt{33}\times\sqrt{14}}\right)\\ & =\cos^{-1}\left(0.2791\right)\\ \theta & =73.8^{\circ}. \end{align*} The angle between $$\vec{a}$$ and $$\vec{b}$$ is $$73.8^{\circ}$$.

2. Find the angle $$\theta$$, between $$\vec{a}\left(1,0,1\right)$$ and $$\vec{b}\left(-2,-1,1\right).$$ \begin{align*} \vec{a}\cdot\vec{b} & =\left(1,0,1\right)\cdot\left(-2,-1,1\right)=-1\\ \mid\vec{a}\mid & =\sqrt{2}\\ \mid\vec{b}\mid & =\sqrt{6}\\ \theta & =\cos^{-1}\left(\frac{\vec{a}\cdot\vec{b}}{\mid\vec{a}\mid\mid\vec{b}\mid}\right)=\cos^{-1}\left(\frac{-1}{\sqrt{2}\times\sqrt{6}}\right)=\cos^{-1}\left(-0.2887\right)\\ \theta & =106.8^{\circ}. \end{align*}

The angle between $$\vec{a}$$ and $$\vec{b}$$ is $$106.8^{\circ}$$.

See Exercises 4 and 5.

Exercise 1

Calculate the dot product of:

$$\text{(a) \left(2,5,-1\right) and \left(4,1,1\right) }$$

$$\text{(b) 3\vec{i} }$$ and $$5\vec{j}$$

$$\text{(c)}$$ $$5\vec{k}$$ and $$\left(\vec{j}+2\vec{k}\right)$$

$$\text{(a) 12}\quad\text{(b) 0\quad\text{(c) 10 } }$$

Exercise 2

Find:

$$\text{(a)\,\left(2,0,4\right)\cdot\left(-3,1,3\right) }$$

$$\text{(b)\,\left(0,5,1\right)\cdot\left(4,0,0\right) }$$

$$\text{(c)\,\left(2\vec{i}+3\vec{k}\right)\cdot\left(7\vec{i}+2\vec{j}+4\vec{k}\right) }$$

$$\text{(a) 6}\quad\text{(b) 0\quad\text{(c) 26 } }$$

Exercise 3

Which of the following vectors are perpendicular?

$$\text{(a)}\,\left(5,2,3\right)$$

$$\text{(b)}\,\left(0,1,-1\right)$$

$$\text{(c)}\,\left(-2,2,2\right)$$

$$\text{(a) and (c), (b) and (c)}$$

Exercise 4

Find the angle between the following pairs of vectors:

$$\text{(a)}\,\left(1,2,3\right)\$$and $$\left(4,-1,0\right)$$

$$\text{(b)}\,\left(2,1,-2\right)\$$and $$\left(1,5,-1\right)$$

$$\text{(c)}\,\left(0,5,1\right)\$$and $$\left(2,0,0\right)$$

$$\text{(d)}\,\left(1,-2,3\right)\$$and $$\left(-4,1,-3\right)$$

$$\text{(e)}\,\left(2,1,-2\right)\$$and $$\left(0,4,0\right)$$

$$\text{(f)}\,\left(0,3,0\right)\$$and $$\left(0,1,0\right)$$

$$\text{(a)\,82.6^{\circ}\quad\text{(b)\,54.7^{\circ}\quad\text{(c)\,90^{\circ}\quad\text{(d)\,141.8^{\circ}\quad\text{(e)\, 70.5^{\circ}\quad\text{(f)\,0^{\circ} } } } } } }$$

Exercise 5

If $$\vec{a}=\left(2,2,2\right),\,\vec{b}=\left(3,2,-1\right),$$ and $$\vec{c}=\left(-1,4,1\right),$$

$$\text{(a)\, }$$Show $$\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}$$

$$\text{(b)\, }$$Rearranging $$\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}$$ gives $$\vec{a}\cdot\left(\vec{b}-\vec{c}\right)=0$$. As $$\vec{b}\neq\vec{c}$$ what is the relationship between $$\vec{a}$$ and $$\left(\vec{b}-\vec{c}\right)$$?

$$\text{(a)\,\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}=8\quad\text{(b) \vec{a} is perpendicular to \left(\vec{b}-\vec{c}\right). } }$$