A vector has a magnitude of 16 and a direction of 120°. How do you write the vector in terms of unit vectors?

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A vector has a magnitude of 16 and a direction of 120°. How do you write the vector in terms of unit vectors?

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Answer 1 · 2017-04-06T05:14:32

$-8 hat i + 8sqrt(3) hat j$

Explanation:

The vector is of the form $a hat i + b hat j, a in RR, b in RR$ . We need to find $a$ and $b$ .

The magnitude of a vector is defined as $sqrt(a^2+b^2)$ .

The direction of a vector can be found using $arctan(b/a)$ .

From the question, we know that $sqrt(a^2+b^2)=16$ and $arctan(b/a)=120^@$ . Since $90^@<120^@<180^@$ , the vector lies in the second quadrant. In other words, $a<0$ and $b>0$ .

From $arctan(b/a)=120^@$ , we know that $b/a=-sqrt(3)$ , or $b=-asqrt(3)$ .

Substitute this into $sqrt(a^2+b^2)=16$ . We get $sqrt(a^2+(-asqrt(3))^2)=16$ . Simplifying this, we get $sqrt(4a^2)=16$ . Solving this gives $a=+-8$ . However, we said before that $a<0$ . Thus, $a=-8$ .

Now, $b=-asqrt(3)=-(-8)sqrt(3)=8sqrt(3)$ .

The vector can be written as $-8 hat i + 8sqrt(3) hat j$ .

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A vector has a magnitude of 16 and a direction of 120°. How do you write the vector in terms of unit vectors?

1 Answer

Explanation:

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