Find a vector \vec a that has the same direction as \langle -6,5,6 \rangle but has length 3.

Question:

Find a vector {eq}\vec a {/eq} that has the same direction as {eq}\langle -6,5,6 \rangle {/eq} but has length {eq}3 {/eq}.

Vector

A quantity that has magnitude as well as direction is known as the vector quantity. The unit vector of any vector is given by the formula {eq}\hat a = \frac{ \vec a}{|\vec a|} {/eq} and the direction of this unit vector is same as the direction of the vector a, but has length 1.

Answer and Explanation:

We have {eq}\vec a=(-6,5,6) {/eq}

The magnitude of this vector is equal to

{eq}= \sqrt{(-6)^{2}+(5)^{2}+(6)^{2}} \\ = \sqrt{36+25+36} \\ = \sqrt{97} {/eq}

And the unit vector of any vector is given by the formula {eq}\hat a = \frac{ \vec a}{|\vec a|} {/eq}

Thus, the unit vector that has the same direction with the given vector is equal to {eq}(\frac{-6}{\sqrt{97}} , \ \frac{5}{\sqrt{97}}, \ \frac{6}{\sqrt{97}}) {/eq}

Now, the vector that has the same direction with the given vector and has a length of 3 units is equal to

{eq}(\frac{-18}{\sqrt{97}} , \ \frac{15}{\sqrt{97}}, \ \frac{18}{\sqrt{97}}). {/eq}


Learn more about this topic:

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Vectors: Definition, Types & Examples

from Common Entrance Test (CET): Study Guide & Syllabus

Chapter 57 / Lesson 3
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