Question

If P(3,4,5), Q(4,6,3), R(-1,2,4), S(1,0,5), then the projection of RS on PQ is

• A. \(-2/3\)
• B. \(-4/3\)
• C. \(1/2\)
• D. 2

Hint:

Projection of \(\vec{a}\) on \(\vec{b}\) = \(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\).

Answer 1

The Correct Option is B

To solve the problem of finding the projection of the vector \(\overrightarrow{RS}\) on the vector \(\overrightarrow{PQ}\), we first need to identify the coordinates of the points and then apply the formulas for vector projection.

1. Identify the coordinates of the points:
   • P: (3, 4, 5)
   • Q: (4, 6, 3)
   • R: (-1, 2, 4)
   • S: (1, 0, 5)
2. Determine the direction vectors:
   • Vector \(\overrightarrow{PQ} = \langle 4-3, 6-4, 3-5 \rangle = \langle 1, 2, -2 \rangle\)
   • Vector \(\overrightarrow{RS} = \langle 1-(-1), 0-2, 5-4 \rangle = \langle 2, -2, 1 \rangle\)
3. Use the formula for the projection of one vector onto another:
   • The projection of a vector \(\mathbf{a}\) on \(\mathbf{b}\) is given by: \(\mathrm{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}\)
   • Compute the dot product \(\overrightarrow{RS} \cdot \overrightarrow{PQ}\):
     • \(\overrightarrow{RS} \cdot \overrightarrow{PQ} = (2)(1) + (-2)(2) + (1)(-2) = 2 - 4 - 2 = -4\)
   • Compute the dot product \(\overrightarrow{PQ} \cdot \overrightarrow{PQ}\):
     • \(\overrightarrow{PQ} \cdot \overrightarrow{PQ} = (1)(1) + (2)(2) + (-2)(-2) = 1 + 4 + 4 = 9\)
4. Calculate the projection of \(\overrightarrow{RS}\) on \(\overrightarrow{PQ}\):
   • \(\mathrm{proj}_{\overrightarrow{PQ}} \overrightarrow{RS} = \frac{-4}{9} \langle 1, 2, -2 \rangle = \langle -\frac{4}{9}, -\frac{8}{9}, \frac{8}{9} \rangle\)
   • The projection vector has a length of magnitude \(-\frac{4}{3}\), noticing that it is a scalar multiple of the unit projection. Thus, the answer is \(-\frac{4}{3}\).
5. Conclusion: The projection of \(\overrightarrow{RS}\) on \(\overrightarrow{PQ}\) is \(-\frac{4}{3}\). Therefore, the correct answer is \(-\frac{4}{3}\).