Download App

Straight line in space — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsStraight line in space3 Marks
Question
Write the value of $\lambda$ for which the lines $\frac{\text{x}-3}{-3}=\frac{\text{y}+2}{2\lambda}=\frac{\text{z}+4}{2}$ and $\frac{\text{x}+1}{3\lambda}=\frac{\text{y}-2}{1}=\frac{\text{z}+6}{-5}$ are perpendicular to each other.
✓

Answer

 We have
$\frac{\text{x}-3}{-3}=\frac{\text{y}+2}{2\lambda}=\frac{\text{z}+4}{2}$
$\frac{\text{x}+1}{3\lambda}=\frac{\text{y}-2}{1}=\frac{\text{z}+6}{-5}$
The given lines are parallel to vector $\vec{\text{b}}_1=-3\hat{\text{i}}+2\lambda\hat{\text{j}}+2\hat{\text{k}}$ and $\vec{\text{b}}_2=3\lambda\hat{\text{i}}+\hat{\text{j}}-5\hat{\text{k}}.$
For $\vec{\text{b}}_1\perp\vec{\text{b}}_2,$ we must have
$\vec{\text{b}}_1.\vec{\text{b}}_2=0$
$\Rightarrow\big(-3\hat{\text{i}}+2\lambda\hat{\text{j}}+2\hat{\text{k}}\big).\big(3\lambda\hat{\text{i}}+\hat{\text{j}}-5\hat{\text{k}}\big)=0$
$\Rightarrow-7\lambda-10=0$
$\Rightarrow\lambda=-\frac{10}7{}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks" from Straight line in space→Straight line in space — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integrals:

$\int\text{e}^{\text{x}}\big(\cot\text{x}-\text{cosec}^2\text{x}\big)\text{dx}$

→
On the set Q of all ration numbers if a binary operation * is defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{5},$ prove that * is associative on Q.
→
Show that the vectores $\vec{\text{a}}=3\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}-3\hat{\text{j}}+5\hat{\text{k}},\vec{\text{c}}=2\hat{\text{i}}+\hat{\text{j}}-4\hat{\text{k}}$from a right-angled triangle.
→
Show that the line through the points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (-1, -2, 1) and, (1, 2, 5).
→
Evaluate the following integrals:
$\int\text{x}^2\cos\text{x dx}$
→
Evaluate the following integrals:
$\int\frac{\text{x}^2\text{dx}}{\text{x}^6-\text{a}^6}\text{dx}$
→
Find the probability that in 10 throws of a fair die, a score which is a multiple of 3 will be obtained in at least 8 of the throws.
→
Write the composition table for the binary operation multiplication modulo 10 (×10) on the set S = {2, 4, 6, 8}.
→
Find the coordinates of the point where the line through (3, -4, -5) and (2, -3, 1) corsses the plane 2x + y + z = 7.
→
A man 2 metres high walks at a uniform speed of 5km/-hr away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases.

 

→