Download App

Straight line in space — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsStraight line in space4 Marks
Question
Show that the three lines with direction cosines $\frac{12}{13},\frac{-3}{13},\frac{-4}{13},\frac{4}{13},\frac{12}{13},\frac{3}{13},\frac{3}{13},\frac{-4}{13},\frac{12}{13}$ are mutually perpendicular.
✓

Answer

let $\text{l}_1=\frac{12}{13},\text{m}_1=-\frac{3}{13},\text{n}_1=-\frac{4}{13}$
$\text{l}_2=\frac{4}{13},\text{m}_2=\frac{12}{13},\text{n}_2=\frac{3}{13}$
$\text{l}_3=\frac{3}{13},\text{m}_3=-\frac{4}{13},\text{n}_3=\frac{12}{13}$
$\text{l}_1\text{l}_2+\text{m}_1\text{m}_2+\text{n}_1\text{n}_2$
$=\frac{12}{13}\times\frac{4}{13}+\big(-\frac{3}{13}\big)\times\frac{12}{13}+\big(-\frac{4}{13}\big)\times\frac{3}{13}$
$=\frac{48-36-13}{169}=0$
$\text{l}_2\text{l}_3+\text{m}_2\text{m}_3+\text{n}_2\text{n}_3$
$=\frac{4}{13}\times\frac{3}{13}+\frac{12}{13}\times\Big(-\frac{4}{13}\Big)+\frac{3}{13}\times\frac{12}{13}$
$=\frac{12-48+36}{169}=0$
$\text{l}_1\text{l}_3+\text{m}_1\text{m}_3+\text{n}_1\text{n}_3$
$=\frac{12}{13}\times\frac{3}{13}+\Big(-\frac{3}{13}\Big)\times\Big(-\frac{4}{13}\Big)+\Big(-\frac{4}{13}\Big)\times\frac{12}{13}$
$=\frac{36+12-48}{169}=0$
$\therefore$ The lines are mutually perpendicular.

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 "4 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

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}=\sin\text{x}\cos\text{x}$
→
A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs. 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs. 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum  of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?
→
If $\cos^{-1}\frac{\text{x}}{2}+\cos^{-1}\frac{\text{y}}{3}=\alpha,$ then prove that 

$9\text{x}^2-12\text{xy}\cos\alpha+4\text{y}^2=36\sin^2\alpha.$

→
Find the points of local maxima or local minima, if any, of the following functions, using the first derivatives test. Also, find the local maximum or local minimum values, as the case may be: 
$\text{f}(\text{x})=\sin2\text{x},0\leq\text{x}\leq\pi$ 
→
Two probability distributions of the discrete random variable X and Y are given below.

$\text{X}$

$0$

$1$

$2$

$3$

$\text{P}(\text{X})$

$\frac{1}{5}$

$\frac{2}{5}$

$\frac{1}{5}$

$\frac{1}{5}$

 

$\text{Y}$

$0$

$1$

$2$

$3$

$\text{P}(\text{Y})$

$\frac{1}{5}$

$\frac{3}{10}$

$\frac{2}{5}$

$\frac{1}{10}$
Prove that E(Y2) = 2E(X).

→
If R and S are relations on a set A, then prove that:
  1. R and S are symmetric $\Rightarrow\ \text{R}\cap\text{S}$ and $\text{R}\cup\text{S}$ are symmetric
  2. R is reflexive and S is any relation $\Rightarrow\ \text{R}\cup\text{S}$ is reflexive.
→
Three persons A, B and C apply for a job of Manager in a Private Company. Chances of their selection (A, B and C) are in the ratio 1 : 2 : 4. The probabilities that A, B and C can introduce changes to improve profits of the company are 0.8, 0.5 and 0.3, respectively. If the change does not take place, find the probability that it is due to the appointment of C.
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\frac{\text{x}^{11}-3\text{x}^9+5\text{x}^7-\text{x}^5+1}{\cos^2\text{x}}\text{ dx}$
→
Two bad eggs are accidently mixed up with ten good ones. Three eggs are drawn at random with replacement from this lot. Compute the mean for the number of bad eggs drawn.
→
Evaluate the following integrals:
$\int\text{x}\sqrt{\text{x}^2+\text{x}}\text{dx}$
→