9Bunitvectors


 * Relative Vectors**

The vector **OP** is the position vector for the point **P**. Similarly **OQ** is the position vector for the point **Q**. Using vector addition, we can get an expression for the vector **PQ**.

math \\ \overrightarrow{OP}+\overrightarrow{PQ}=\overrightarrow{OQ} \quad \textit{ subtract } \overrightarrow{OP} \textit{ from both sides} \qquad .\\. \\ \overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP} math


 * PQ** is sometimes called a **relative vector**.

In other words: Any vector can be found using the position vectors of its endpoints = (position vector of HEAD) – (position vector of TAIL)

Example

Find the vector **AB**, from **A**(2, 3) to **B**(–3, 4)

math \\ \overrightarrow{OA}=2\underline{i}+3\underline{j} \qquad \\. \\ \overrightarrow{OB}=-3\underline{i}+4\underline{j} math

math \\ \overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA} \qquad. \\ . \\ . \quad = \big( -3\underline{i}+4\underline{j} \big) - \big( 2\underline{i}+3\underline{j} \big) \qquad. \\ . \\ . \quad = -5\underline{i}+\underline{j} math

**Unit Vectors**

A **unit vector** is a vector with a magnitude of one.

Remember **__i__** is a unit vector in the direction of the x-axis.

We can make a unit vector in the direction of any vector, **__v__**.

To indicate a unit vector in the direction of **__v__**, we put a cap (^) on top of it.

To find the unit vector in the direction of **__v__**, divide it by the magnitude, **|__v__|**. math \underline{\hat{v}}=\dfrac{\underline{v}}{|\underline{v}|}=\dfrac{a\underline{i}+b\underline{j}}{\sqrt{a^2+b^2}} \qquad. math

Example

Find a unit vector in the direction of **__v__** = 3**__i__** + 4**__j__**

math \underline{\hat{v}}=\dfrac{3\underline{i}+4\underline{j}}{\sqrt{3^2+4^2}}=\dfrac{3\underline{i}+4\underline{j}}{5} =\frac{3}{5}\underline{i}+\frac{4}{5}\underline{j} \qquad. math

math \\ \lbrace \textit{Notice that the magnitude of } \underline{\hat{v}}=\frac{3}{5}\underline{i}+\frac{4}{5}\underline{j} \textit{ is one} \rbrace \qquad. \\ . \\ math
 * \underline{\hat{v}}|=\sqrt{ \big(\frac{3}{5} \big)^2 + \big( \frac{4}{5} \big)^2} = \sqrt{\frac{9}{25}+\frac{16}{25}}=1

Your calculator will find a unit vector in 2 or 3 dimensions Use the unitV function in the ACTION menu, VECTOR submenu.

Finding the Angle a Vector Makes with the axes
The unit vector values are the cosine of the angle the vector makes with that axis.

math \underline{\hat{v}} = \cos ( \alpha ) \underline {i} + \cos ( \beta ) \underline {j} + \cos ( \gamma ) \underline {k} \qquad. \\.\\ \text {where: } \quad \; \alpha = \text{ angle between v and x-axis} \qquad. \\ .\qquad \qquad \beta = \text { angle between v and y-axis} \\ . \qquad \qquad \gamma = \text { angle between v and z-axis} math

Vectors on the Classpad

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