Property

Details

Definition

$\mathbf{A}·\mathbf{B}\=\sum _{k\=1}^n{a}_k{b}_k$

Commutation

$\mathbf{A}·\mathbf{B}\=\mathbf{B}·\mathbf{A}$

Norm of A

$\mathbf{A}·\mathbf{A}\=\sum _{k\=1}^n{a}_k^2\={∥\mathbf{A}∥}^2$

Distance between

Points A and B

$∥\mathbf{A}\mathbf{-}\mathbf{B}∥$ or $∥\mathbf{B}\mathbf{-}\mathbf{A}∥$, where A and B are position vectors to points A and B$$

Alternate Definition

$\mathbf{A}·\mathbf{B}\=∥\mathbf{A}∥∥\mathbf{B}∥\cos \(\mathrm{\θ}\)$, where $\mathrm{\θ}$ is the angle between A and B

Angle between Vectors

$\cos \left(\mathrm{\θ}\right)\=\frac{\mathbf{A}·\mathbf{B}}{∥\mathbf{A}∥ ∥\mathbf{B}∥}$

Orthogonality of A and B

$\mathbf{A}·\mathbf{B}\=0\Rightarrow$A ⊥ B, provided neither A nor B is the zero vector

Direction Cosines for A

$\cos \left(\mathrm{\α}\right)\=\frac{\mathbf{A}·\mathbf{i}}{∥\mathbf{A}∥}$ = $\frac{a_1}{∥\mathbf{A}∥}$ 

$\cos \left(\mathrm{\β}\right)\=\frac{\mathbf{A}\mathbf{·}\mathbf{j}}{∥\mathbf{A}∥}$ = $\frac{a_2}{∥\mathbf{A}∥}$ 

$\cos \left(\mathrm{\γ}\right)\=\frac{\mathbf{A}\mathbf{·}\mathbf{k}}{∥\mathbf{A}∥}$ = $\frac{a_3}{∥\mathbf{A}∥}$ 

Scalar Projection of B on A

$∥{\mathbf{B}}_{\mathbf{A}}∥$ =  $\frac{\mathbf{B}·\mathbf{A}}{∥\mathbf{A}∥}$

Vector Projection of B on A

${\mathbf{B}}_{\mathbf{A}}\=\left(\frac{\mathbf{B}·\mathbf{A}}{∥\mathbf{A}∥}\right)\frac{\mathbf{A}}{∥\mathbf{A}∥}$ = $\frac{\mathbf{B}·\mathbf{A}}{\mathbf{A}·\mathbf{A}}\mathbf{A}$

Component of B orthogonal to A

${\mathbf{B}}_{\perp \mathbf{A}}\=\mathbf{B}-{\mathbf{B}}_{\mathbf{A}}$

Product Rule of Differentiation

$\left(\mathbf{A}·\mathbf{B}\right)\prime \=\mathbf{A}·\mathbf{B}\prime \+\mathbf{B}·\mathbf{A}\prime$

Table 1.3.1   The dot product, its definition, properties, and consequences

Maple State

DotProduct 
(Command)

Math Mode

Underlying

Field

Top-Level

No

Period or $·$ from Common Symbols palette

Complex

Student MultivariateCalculus

Yes

Period or $·$ from Common Symbols palette

Real

Student LinearAlgebra

No

Period or $·$ from Common Symbols palette

Real

LinearAlgebra

Yes

Period or $·$ from Common Symbols palette

Complex

Student VectorCalculus

Yes

Period or $·$ from Common Symbols palette

Real

VectorCalculus

Yes

Period or $·$ from Common Symbols palette

Real

Table 1.3.2   Implementing the dot product in Maple

Example 1.3.1

If $\mathbf{A}\=3 \mathbf{i}\+7 \mathbf{j}$ and $\mathbf{B}\=-4 \mathbf{i}\+5 \mathbf{j}$,

Example 1.3.2

If $\mathbf{A}\=3 \mathbf{i}\+2 \mathbf{j}\+7 \mathbf{k}$ and $\mathbf{B}\=4 \mathbf{i}-5 \mathbf{j}\+6\mathbf{ }\mathbf{k}$,

Example 1.3.3

Using the law of cosines, verify the equivalence of $\mathbf{A}·\mathbf{B}$ and $∥\mathbf{A}∥∥\mathbf{B}∥\cos \(\mathrm{\theta }\)$

Example 1.3.4

In the $\mathrm{xy}$-plane, obtain all vectors that are orthogonal to $\mathbf{A}\=3 \mathbf{i}\+2 \mathbf{j}$.

Example 1.3.5

If $\mathbf{A}\=3 \mathbf{i}\+2 \mathbf{j}\+5 \mathbf{k}$ and $\mathbf{B}\=-4 \mathbf{i}\+5 \mathbf{j}\+\mathrm{\λ} \mathbf{k}$, find all values of  $\mathrm{\λ}$ that make B orthogonal to A.

Example 1.3.6

Use vector methods to find the distance between the points A:$\left(4\,5\right)$ and B:$\left(-3\,2\right)$.

Example 1.3.7

Use vector methods to find the distance between the points A:$\left(4\,5\,7\right)$ and B:$\left(-3\,2\,3\right)$.

Example 1.3.8

Suppose the components of the planar vectors A and B are functions of $t$, and the derivative of such vectors is defined to be the vector of componentwise derivatives. If the prime denotes differentiation with respect to $t$, show that

Example 1.3.9

Example 1.3.10

Find a vector of unit length that is orthogonal to the vectors $\mathbf{A}\=\mathbf{i}\+\mathbf{j}\+\mathbf{k}$ and $\mathbf{B}\=2 \mathbf{i}\+3 \mathbf{j}-\mathbf{k}$.

Example 1.3.11

Show that $\mathbf{A}·\mathbf{A}\=0$ is a necessary and sufficient condition for A to be the zero vector.

Example 1.3.12

Use vector methods to prove that an angle inscribed in a semicircle is necessarily a right angle.

Example 1.3.13

Do the terminal points of the position vectors to points A:$\left(2\,-3\,7\right)$, B:$\left(1\,-1\,10\right)$,

and C:$\left(3\,-5\,4\right)$ lie on a straight line?

Example 1.3.14

If A and B are unit vectors with $\mathrm{\θ}$ the angle between them, show that $\sin \left(\mathrm{\θ}\/2\right)\=\frac{1}{2}∥\mathbf{A}-\mathbf{B}∥$.

Example 1.3.15

If $\mathbf{A}\=a\left(x\right) \mathbf{i}\+b\left(x\right) \mathbf{j}$ and $\mathbf{B}\=u\left(x\right) \mathbf{i}\+v\left(x\right) \mathbf{j}$, verify the product rule for differentiating the dot product $\mathbf{A}·\mathbf{B}$.