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Chapter 6: Problem 89

If vector \(\mathbf{v}\) is represented by an arrow, how is \(-3 \mathbf{v}\) represented?

Short Answer

Expert verified
The vector \(-3 \mathbf{v}\) is represented by an arrow three times as long as \(\mathbf{v}\), pointing in the exact opposite direction.

Step by step solution

01

Understanding the Vector \(\mathbf{v}\)

In a graphical representation, the vector \(\mathbf{v}\) is usually depicted as an arrow pointing from the origin to a point in space. The length of the arrow denotes the magnitude of the vector, and the direction of the arrow indicates the direction of the vector.
02

Scaling the Vector

Scaling a vector involves changing its magnitude without altering its direction. In this case, the scaling factor is \(-3\), which would make the magnitude of the new vector three times the original. However, the negative sign indicates there's an additional operation.
03

Understanding the Negative Sign with the Scaling

The negative sign in front of the scalar indicates not just the scaling of the vector, but also a reversal of its direction. Graphically, this means that if the arrow of \(\mathbf{v}\) was pointing towards the north, the arrow for \(-3 \mathbf{v}\) will be pointing towards the south and it will be three times as long as the arrow for \(\mathbf{v}\).

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Most popular questions from this chapter

Describe a test for symmetry with respect to the line \(\theta=\frac{\pi}{2}\) in which \(r\) is not replaced.

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=3 \mathbf{i}, \quad \mathbf{w}=-4 \mathbf{j}$$

If you are given two sides of a triangle and their included angle, you can find the triangle's area. Can the Law of Sines be used to solve the triangle with this given information? Explain your answer.

Find the angle, in degrees, between \(\mathbf{v}\) and \(\mathbf{w}.\) $$\mathbf{v}=2 \cos \frac{4 \pi}{3} \mathbf{i}+2 \sin \frac{4 \pi}{3} \mathbf{j}, \quad \mathbf{w}=3 \cos \frac{3 \pi}{2} \mathbf{i}+3 \sin \frac{3 \pi}{2} \mathbf{j}$$

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=\mathbf{i}+\mathbf{j}, \quad \mathbf{w}=-\mathbf{i}+\mathbf{j}$$
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