Find the angle between the plane: 2x - y + z = 4 and x + y + 2z =… - Vidyadip

Question

Find the angle between the plane:
2x - y + z = 4 and x + y + 2z = 3

✓

Answer

We know that the angle between the planes,
$\text{a}_1\text{x}+\text{b}_1\text{y}+\text{c}_1\text{z}+\text{d}_1=0$ and $\text{a}_2\text{x}+\text{b}_2\text{y}+\text{c}_2\text{z}+\text{d}_2=0$ is given by
$\cos\theta=\frac{\text{a}_1\text{a}_2+\text{b}_1\text{b}_2+\text{c}_1\text{c}_2}{\sqrt{\text{a}_1^2+\text{b}_1^2+\text{c}_1^2}\sqrt{\text{a}_2^2+\text{b}_2^2+\text{c}^2_2}}$
So, the angle between 2x - y + z = 4 and x + y + 2z = 3 is given by
$\cos\theta=\frac{(2)(1)+(-1)(1)+(1)(2)}{\sqrt{2^2+(-1)^2+1^2}\sqrt{1^2+1^2+1^2}}$
$=\frac{2-1+2}{\sqrt{4+1+1}\sqrt{1+1+4}}=\frac{3}{\sqrt{6}\sqrt{6}}$
$=\frac{3}{6}=\frac{1}{2}$
$\theta=\cos^{-1}\Big(\frac{1}{2}\Big)=\frac{\pi}{3}$

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 Question" from The plane→The plane — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the vector equation of the plane with intercepts 3, -4 and 2 on x, y and z-axis respectively.

Evaluate the following integrals:
$\int\frac{\cos\text{x}-\sin\text{x}}{1+\sin2\text{x}}\text{dx}$

Find the second-order derivative of the function log(log x)

Find the angle between the following pair of lines:

$\frac{\text{x}}{2}=\frac{\text{y}}{2}=\frac{\text{z}}{1}\ \text{and}\ \frac{\text{x}-5}{4}=\frac{\text{y}-2}{1}=\frac{\text{z}-3}{8}$

Find mean $u,$variance $\sigma^{2}$for the following probability distribution:

X	0	1	2	3
P(X)	$\frac{1}{8}$	$\frac{3}{8}$	$\frac{3}{8}$	$\frac{1}{8}$

If the mean and variance of a binomial distribution are 4 and 3, respectively, find the probability of no success.

An urn contains 3 white, 4 red and 5 black balls. Two balls are drawn one by one without replacement. What is the probability that at least one ball is black?

Evaluate the following integrals:
$\int\frac{1}{\sqrt{\tan^{-1}\text{x}}.(1+\text{x}^2)}\text{dx}$

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\sin^2\text{y}$

Find the angle between the lines whose direction ratios are a, b, c and b - c, c - a, a - b.