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Wize University Calculus 3 Textbook > Vectors & Geometry of Space

Dot Product

Properties of Vectors, Length of Vectors, and Unit VectorsPrevious Section
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Dot Product

Given two vectors, a⃗=<a1, a2, a3> \vec{a}=<a_1,~a_2,~a_3>~and b⃗=<b1, b2, b3> \vec{b}=<b_1,~b_2,~b_3>~, then the dot product is:
a⃗⋅b⃗=a1b1+a2b2+a3b3ora⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\begin{array}{rcl} \vec{a}\cdot\vec{b}&=&{a_1b_1}+{a_2b_2}+{a_3b_3}&&\text{\color{magenta}or}\\\\ \vec{a}\cdot\vec{b}&=&|\vec{a}||\vec{b}|\cos\theta \end{array}

Remember the dot product returns a scalar (i.e. a number). The dot product is also known as the scalar/inner product.

Geometric Interpretation of the Dot Product

A dot product is the magnitude of one vector time the portion of the other vector that points in the same direction as that vector.

Projections

We can also look at the dot production in terms of projections, where the projection is labeled below:


The vector projection of a⃗\vec{a} onto b⃗\vec{b} is defined as:

projb⃗a⃗=a⃗⋅b⃗∣∣b⃗∣∣2⋅b⃗=a⃗⋅b⃗∣b∣⋅b⃗∣b∣\begin{array}{rcl} proj_{\vec{b}}\vec{a}&=&\dfrac{\vec{a}\cdot{}\vec{b}}{||\vec{b}||^2}\cdot{}\vec{b}\\\\ &=&\dfrac{\vec{a}\cdot{}\vec{b}}{|b|}\cdot{}\dfrac{\vec{b}}{|b|} \end{array}

The scalar projection of a⃗\vec{a} onto b⃗\vec{b} is defined as:

Compb⃗a⃗=a⃗⋅b⃗∣b⃗∣Comp_{\vec{b}}\vec{a}=\dfrac{\vec a\cdot\vec b}{|\vec{b}|}

where Compb⃗a⃗=∣a∣cos⁡θComp_{\vec{b}}\vec{a}=|a|\cos\theta

Properties

1.u⃗⋅(v⃗+w⃗)=u⃗⋅v⃗+u⃗⋅w⃗2.(c⋅v⃗)⋅w⃗=v⃗⋅(c⋅w⃗)=c⋅(v⃗⋅w⃗)3.v⃗⋅w⃗=w⃗⋅v⃗4.v⃗⋅0⃗=05.v⃗⋅v⃗=∣∣v⃗∣∣26.If  v⃗⋅v⃗=0then, v⃗=0\begin{array}{crcl} 1.&\vec{u}\cdot(\vec{v}+\vec{w})&=&\vec{u}\cdot\vec{v}+\vec{u}\cdot\vec{w}\\\\ 2.&(c\cdot\vec{v})\cdot\vec{w}&=&\vec{v}\cdot(c\cdot\vec{w})\\&=&c\cdot(\vec{v}\cdot\vec{w})\\\\ 3.&\vec{v}\cdot\vec{w}&=&\vec{w}\cdot\vec{v}\\\\ 4.&\vec{v}\cdot\vec{0}&=&0\\\\ 5.&\vec{v}\cdot\vec{v}&=&||\vec{v}||^2\\\\ 6.&\text{If}~~\vec{v}\cdot{}\vec{v}&=&0\\ &\text{then},~\vec{v}&=&0 \end{array}


Write it Down
If u⃗⋅v⃗>0\vec{u}\cdot\vec{v}>0, then the angle created by u⃗ \vec{u}~and v⃗ \vec{v}~is acute.

If u⃗⋅v⃗<0, \vec{u}\cdot\vec{v}<0,~then the angle created by u⃗ \vec{u}~and v⃗ \vec{v}~ is obtuse.

If u⃗⋅v⃗=0, \vec{u}\cdot\vec{v}=0,~then the angle created by u⃗ \vec{u}~and v⃗ \vec{v}~is θ=π2\theta=\frac{\pi}{2}, indicating that u⃗ & v⃗\vec{u}~\&~\vec{v} are orthogonal to each other.


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Example

Calculate the angle between vectors a⃗=⟨1,4,−2⟩\vec{a}=\langle 1,4,-2\rangle and b⃗=<2,2,−1> \vec{b}=<2,2,-1>~(leave the answer in exact form)

To calculate the angle, we will use:
cos⁡(θ)=a⃗⋅b⃗∣a∣∣b∣\cos(\theta)=\dfrac{\vec{a}\cdot\vec{b}}{|a||b|}
cos⁡θ=(2+8+212+42+(−2)222+22+(−1)2)=12219=421\cos\theta=\Bigg(\dfrac{2+8+2}{\sqrt{1^2+4^2+(-2)^2}\sqrt{2^2+2^2+(-1)^2}}\Bigg)=\dfrac{12}{\sqrt{21}\sqrt{9}}=\dfrac{4}{\sqrt{21}}


θ=cos⁡−1(421)\theta=\cos^{-1}\left(\dfrac{4}{\sqrt{21}}\right)

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Example

Find the scalar and vector projection of b⃗=⟨2,1,3⟩\vec{b}=\langle 2,1,3\rangle onto a⃗=⟨−1,4,2⟩\vec{a}=\langle -1,4,2\rangle.

Compab⃗=a⃗⋅b⃗∣a∣=−2+4+6(−1)2+42+22=821\textrm{Comp}_a{\vec{b}}=\dfrac{\vec{a}\cdot\vec{b}}{|a|} =\dfrac{-2+4+6}{\sqrt{(-1)^2+4^2+2^2}}=\dfrac{8}{\sqrt{21}}

Projab⃗=a⃗⋅b⃗∣a∣⋅a⃗∣a∣=821<−1, 4, 2>\textrm{Proj}_a\vec{b}=\dfrac{\vec{a}\cdot\vec{b}}{|a|}\cdot \dfrac{\vec{a}}{|a|}=\dfrac{8}{21}\rowvecth{-1}{4}{2}

Practice

Let a⃗= <1, 2, 0> and b⃗= <2, −1, 10> \vec{a}=~<1,~2,~0>~\text{and}~\vec{b}=~<2,~-1,~10>~ be two vectors in space.

Do a⃗ and b⃗ \vec{a}~\text{and}~\vec{b}~form a rectangle, R, or a square, S? Provide the coordinates of the shape, R or S.