The direction cosines of the line passing through the point P 1-2-5 and Q -1-2-3 isa 1-21- 2-21- 4-21b -1-21-2-21-4-21c -1-21-2-21- 4-21d 1-21- 2-21-4-21

The direction cosines of a line passing through the points P(x_1, y_1, z_1) and Q(x_2, y_2, z_2) are x_2 x_1PQ,y_2 y_1PQ,z_2 z_1PQ, where PQ=√((x_2 x_1)^2+(y_2 ...

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Standard XII

Mathematics

Direction Cosines

The direction...

Question

The direction cosines of the line passing through the point $P (1, 2, - 5)$ and $Q (- 1, - 2, 3)$ is

$(a) (\frac{1}{\sqrt{21}}, \frac{2}{\sqrt{21}}, \frac{4}{\sqrt{21}}) (b) (- \frac{1}{\sqrt{21}}, - \frac{2}{\sqrt{21}}, - \frac{4}{\sqrt{21}})$
$(c) (- \frac{1}{\sqrt{21}}, - \frac{2}{\sqrt{21}}, \frac{4}{\sqrt{21}}) (d) (\frac{1}{\sqrt{21}}, \frac{2}{\sqrt{21}}, - \frac{4}{\sqrt{21}})$

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Solution

The direction cosines of a line passing through the points $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$ are $\frac{x_{2} - x_{1}}{P Q}, \frac{y_{2} - y_{1}}{P Q}, \frac{z_{2} - z_{1}}{P Q}$ , where $P Q = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}}$

Given, $P (1, 2, - 5)$ and $Q (- 1, - 2, 3)$
$P Q = \sqrt{(4 + 16 + 64)} = 2 \sqrt{21}$

Hence D.C's are $(- \frac{1}{\sqrt{21}}, - \frac{2}{\sqrt{21}}, \frac{4}{\sqrt{21}})$
Ans : $(c)$

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The direction cosines of the line passing through the point P(1,2,−5) and Q(–1,−2,3) is (a) (1√21,2√21,4√21)(b) (−1√21,−2√21,−4√21) (c) (−1√21,−2√21,4√21)(d) (1√21,2√21,−4√21)

The direction cosines of a line passing through the points P(x1,y1,z1) and Q(x2,y2,z2) are x2−x1PQ,y2−y1PQ,z2−z1PQ, where PQ=√(x2−x1)2+(y2−y1)2+(z2−z1)2 Given, P(1,2,−5) and Q(–1,−2,3) PQ=√(4+16+64)=2√21 Hence D.C's are (−1√21,−2√21,4√21) Ans : (c)