12.3 The Dot Product

The dot product is a way to multiply vectors. Ends up with a real number called a scalar.

If $\vec{a}=<a_{1}, a_{2}, a_{3}>$ and $\vec{b}=<b_{1}, b_{2}, b_{3}>$, then the dot product of $\vec{a}$ and $\vec{b}$ is the number $\vec{a}\cdot\vec{b}$ given by: $\vec{a}\cdot\vec{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}, b_{3}$ Dot proucts are also called "scalar product" or "inner product".

Properties of Dot Product

1. $\vec{a}\cdot\vec{a}=|\vec{a}|^2$ because the magnitude is $|\vec{a}|=\sqrt{a_{1}^2+a_{2}^2}$ and $\vec{a}\cdot\vec{a}=a_{1}^2+a_{2}^2$
2. $\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}$
3. $\vec{a}\cdot(\vec{b}+\vec{c})= \vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}$
4. $(c\vec{a})\cdot\vec{b}=c(\vec{a}\cdot\vec{b})=\vec{a}\cdot(c\vec{b})$

Uses of Dot Product

$\vec{a}\cdot\vec{b}$ can be given a geometric interpretation in terms of the angle $\theta$ between $\vec{a}$ and $\vec{b}$.

Theorem (proof uses law of cosine): If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then $\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos{\theta}$ Two vectors $\vec{a}$ and $\vec{b}$ are orthogonal (perpendicular) if and only if $\vec{a}\cdot\vec{b}=0$. If $\vec{a}\cdot\vec{b}\neq0$, we know that they are not perpendicular.

The sign of $\vec{a}\cdot\vec{b}$ is reliant on the $\cos{\theta}$ because $|\vec{a}||\vec{b}|$ are always positive. Therefore, the dot product is:

• positive if $\vec{a}$ and $\vec{b}$ point in the same general direction. $0<\theta<\frac{\pi}{2}$
• $0$ if perpendicular
• negative if they point in generally opposite directions. $\frac{\pi}{2}<\theta<\pi$
• $|\vec{a}||\vec{b}|$ if they point in the same direction.
• -$|\vec{a}||\vec{b}|$ if they point in opposite directions.

Projection

Vector Projection

Two vectors $\vec{a}$ and $\vec{b}$ have the same initial point P. $proj_{a}\vec{b}$ represents the vector projection (distance and direction) of $\vec{b}$ onto $\vec{a}$ (The shadow of $\vec{b}$). Read as "prorjecting onto $\vec{a}$, the vector of $\vec{b}$".

Vector projection represented by $proj_{a}\vec{b}=(\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|})\frac{\vec{a}}{|\vec{a}|}=\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|^2}\vec{a}$ because it's the unit vector ($\frac{\vec{a}}{|\vec{a}|}$) in the direction of $\vec{a}$ multiplied by the scalar projection.

Scalar Projection

The scalar projection (magnitude) of $\vec{b}$ onto $\vec{a}$ ("component of $\vec{b}$ along $\vec{a}$" or $comp_{a}\vec{b}$) is defined to be the signed magnitude of the vector projecetion.

The scalar projection can be calculated by $comp_{a}\vec{b}=\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|}$ because $comp_{a}\vec{b}=|\vec{b}|\cos{\theta}$. Replace $\cos{\theta}$ with $\cos{\theta}=\frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$ which comes from $\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos{\theta}$

Work and The Dot Product

In physics, work is defined as $W=Fd$. Only includes the force directed along the line of motion of the object. Suppose $\vec{D}$ is the displacement vector. Work done is defined to be the $comp_{d}\vec{F}$ multiplied by $|\vec{D}|$ (distance moved) or the dot product of $\vec{F}\cdot\vec{D}$ $W=(|\vec{F}|\cos{\theta})|\vec{D}|=\vec{F}\cdot\vec{D}$