9Cdotproduct

There are two ways to multiply vectors together.
 * Multiplying Two Vectors**

The ** dot product ** gives a __scalar__ answer. math . \qquad \underline{u} \centerdot \underline{v} \qquad. math

The ** cross product ** gives a __vector__ answer. math . \qquad \underline{u} \times \underline{v} \qquad. math

We only study the scalar product (dot product) in this course.

Dot Product (Scalar Product) math \text{If } \underline{u}=a\underline{i}+b\underline{j} \text{ and } \underline{v}=c\underline{i}+d\underline{j} \qquad. math

then the dot product is defined as math . \qquad \underline{u} \centerdot \underline{v} = ac+bd \qquad. math

ie the product of the **__i__** coefficients plus the product of the **__j__** coefficients {plus the product of the **__k__** coefficients} Example 1 Find the scalar product of math . \qquad \underline{u}=3\underline{i}+4\underline{j} \text{ and } \underline{v}=2\underline{i}-2\underline{j} \qquad. math

math . \qquad \underline{u} \centerdot \underline{v} = \big( 3 \times 2 \big) + \big( 4 \times -2 \big) = -2 \qquad \lbrace \textit{No units} \rbrace \qquad. math

On the calculator: (See Vectors on the Classpad) the **dotP** function is in the Action/Vector menu


 * dotP([3 4], [2 -2] )**

2nd Definition of Dot Product The scalar product (or dot product) is also defined as: math . \qquad \underline{u} \centerdot \underline{v} = uv \cos \theta \qquad. math where theta is the angle between the 2 vectors

This definition can be rearranged to find the angle between the two vectors: math . \qquad \cos \theta = \dfrac{ \underline{u} \centerdot \underline{v} }{ uv } \qquad. math

NOTE math \\ \text{If } \, \theta = 90^{\circ} \; \text{ then } \; \underline{u} \centerdot \underline{v} = 0 \qquad. \\ . \\ \text{If } \, \theta = 0^{\circ} \; \text{ then } \; \underline{u} \centerdot \underline{v} = uv math

From these, we get: math \\ \underline{i} \centerdot \underline{i} = 1 \qquad \underline{j} \centerdot \underline{j} = 1 \qquad \underline{k} \centerdot \underline{k} = 1 \qquad. \\ . \\ \underline{i} \centerdot \underline{j} = 0 \qquad \underline{i} \centerdot \underline{k} = 0 \qquad \textit{etc} math

We also get this very useful rule: ... ... ... Two vectors are perpendicular if their dot product is zero. Example 2

Find the angle between the two vectors: (in degrees to 1 decimal place) math . \qquad \underline{u}=3\underline{i}+4\underline{j} \text{ and } \qquad. \\ . \\ . \qquad\underline{v}=2\underline{i}-2\underline{j} math

First find the dot product: math . \qquad \underline{u} \centerdot \underline{v} = \big( 3 \times 2 \big) + \big( 4 \times -2 \big) = -2 \qquad. math

Then find the magnitude of each vector math \\ . \qquad u=|\underline{u}|=\sqrt{3^2+4^2}=5 \qquad. \\ . \\ . \qquad v=|\underline{v}|=\sqrt{2^2+(-2)^2}=2\sqrt{2} math

Finally use the 2nd definition to find theta: math \\ . \qquad \cos \theta = \dfrac{ \underline{u} \centerdot \underline{v} }{ uv } = \dfrac{ -2 }{ 5 \times 2\sqrt{2} } = \dfrac{ -1 }{ 5\sqrt{2} } \qquad. \\ . \\ . \qquad \theta \approx 98.1^{\circ} math

Note: The other way to find the angle between two vectors would be to find the angle from the positive x-axis for each vector then subtract the two angles. .