Let P be the point (1, −2, 3) and Q be the point (3, 4, 0). Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Length of the projection of PQ on | Value | ||
| (A) | the x-axis | (I) | 3 |
| (B) | the y-axis | (II) | \(\displaystyle\frac{4}{7}\) |
| (C) | the z-axis | (III) | 2 |
| (D) | a line with direction ratios 2, 3, 6 | (IV) | 6 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
The projection of PQ on a line with direction cosines l, m, n has length \(|(x_2-x_1)l+(y_2-y_1)m+(z_2-z_1)n|\). Here
\(x_2-x_1 = 2,\quad y_2-y_1 = 6,\quad z_2-z_1 = -3\)
(A) the x-axis
This matches (III) 2. The x-axis has direction cosines 1, 0, 0, so the projection is \(|2| = 2\).
(B) the y-axis
This matches (IV) 6. With 0, 1, 0 the projection is \(|6| = 6\).
(C) the z-axis
This matches (I) 3. With 0, 0, 1 the projection is \(|-3| = 3\).
(D) a line with direction ratios 2, 3, 6
This matches (II) \(\frac{4}{7}\). The direction cosines are \(\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\):
\( \displaystyle \left|\frac{2(2)+6(3)+(-3)(6)}{7}\right| = \frac{4}{7} \)