Question

Figure 1.55 shows several arrows representing vectors in the xy plane. (a) Which vectors have magnitudes equal to the magnitude of $\overline{\mathit{a}}$? (b) Which vectors are equal to $\overline{\mathit{a}}$?

Short answer

(a) Vectors $\stackrel{\rightharpoonup }{c},\stackrel{\rightharpoonup }{e}$and $\stackrel{\rightharpoonup }{f}$ have the same magnitude as .

(b) Vectors $\stackrel{\rightharpoonup }{c}$and $\stackrel{\rightharpoonup }{f}$are equal to .

Step-by-step solution

Step 1: Magnitude and direction of vectors

While comparing the magnitude of vectors, one can compare the length of the rays.

To check the equality of vectors, both the direction and length of rays must be the same.

Finding vectors whose magnitude is the same as a⇀

(a)

By looking at the figure, it is clear that vectors$\stackrel{\rightharpoonup }{c},\stackrel{\rightharpoonup }{e}$ and$\stackrel{\rightharpoonup }{f}$have the same magnitude as vector$\stackrel{\rightharpoonup }{a}$. This is because the three rays representing these vectors have the same length as the vectorrole="math" localid="1653919248162" $\stackrel{\rightharpoonup }{\mathrm{a}}$.

Therefore, the vectors$\stackrel{\rightharpoonup }{c},\stackrel{\rightharpoonup }{e}$and$\stackrel{\rightharpoonup }{f}$have the same magnitude as the vector$\stackrel{\rightharpoonup }{a}$.

Finding vectors equal to  a⇀

(b)

It is evident from the figure that vectors$\stackrel{\rightharpoonup }{c}$and$\stackrel{\rightharpoonup }{f}$have the same direction and magnitude as vector$\stackrel{\rightharpoonup }{a}$. So, both these are equal to vector$\stackrel{\rightharpoonup }{a}$.

Therefore, the vectors $\stackrel{\rightharpoonup }{c},\stackrel{\rightharpoonup }{f}$ are equal to the vector $\stackrel{\rightharpoonup }{a}$.