Question

In Exercises 57-60, determine which of the vectors is (are) parallel to \(\mathrm{z}\). Use a graphing utility to confirm your results. \(\mathrm{z}=\langle 3,2,-5\rangle\) (a) \langle-6,-4,10\rangle (b) \(\left\langle 2, \frac{4}{3},-\frac{10}{3}\right\rangle\) (c) \langle 6,4,10\rangle (d) \langle 1,-4,2\rangle

Short answer

The vectors (a) \( \langle-6,-4,10\rangle \) and (b) \( \langle 2, \frac{4}{3},-\frac{10}{3} \rangle \) are parallel to the given vector \( \mathrm{z} \). This is because the ratios of corresponding components in these vectors with vector \( \mathrm{z} \) are equal. Thus, these vectors are scalar multiples of the given vector \( \mathrm{z} \). The vectors (c) and (d) are not parallel to \( \mathrm{z} \) as their component ratios with \( \mathrm{z} \) are not equal.

Step-by-step solution

Identify the Given Vector

Identify the given vector \( \mathrm{z} \) which is \( \langle 3,2,-5\rangle \). This is the vector we will compare the other options (a, b, c and d) with to check if they are parallel.

Compare with Vector (a)

Check if the vector (a) \( \langle-6,-4,10\rangle \) is a scalar multiple of the given vector \( \mathrm{z} \). This implies checking if the ratios \( \frac{3}{-6} \), \( \frac{2}{-4} \), and \( \frac{-5}{10} \) are the same. These ratios are the scalars by which the vectors are multiplied. If these ratios are equal, then the vectors are parallel.

Compare with Vector (b)

Check if the vector (b) \( \langle 2, \frac{4}{3},-\frac{10}{3} \rangle \) is a scalar multiple of the given vector \( \mathrm{z} \). This implies checking if the ratios \( \frac{3}{2} \), \( \frac{2}{\frac{4}{3}} \), and \( \frac{-5}{-\frac{10}{3}} \) are the same. If these ratios are equal, then the vectors are parallel.

Compare with Vector (c)

Check if the vector (c) \( \langle 6,4,10\rangle \) is a scalar multiple of the given vector \( \mathrm{z} \). This implies checking if the ratios \( \frac{3}{6} \), \( \frac{2}{4} \), and \( \frac{-5}{10} \) are the same. If these ratios are equal, then the vectors are parallel.

Compare with Vector (d)

Check if the vector (d) \( \langle 1,-4,2\rangle \) is a scalar multiple of the given vector \( \mathrm{z} \). This implies checking if the ratios \( \frac{3}{1} \), \( \frac{2}{-4} \), and \( \frac{-5}{2} \) are the same. If these ratios are equal, then the vectors are parallel.