Find the angle between the planes \(\vec{r}.(4\hat{i}+\hat{j}-2\hat{k})\)=6 and \(\vec{r}.(5\hat{i}-6\hat{j}+\hat{k})\)=7?

\(cos^{-1}⁡\frac{12}{\sqrt{1302}}\)

\(cos^{-1}⁡\frac{1}{\sqrt{1392}}\)

\(cos^{-1}\frac{⁡23}{\sqrt{102}}\)

\(cos^{-1}⁡\frac{15}{\sqrt{134}}\)

If two vectors \(\vec{r}.\vec{n_1}=d_1\) and \(\vec{r}.\vec{n_2}=d_2\) are such that \(\vec{n_1}.\vec{n_2}\)=0, then which of the following is true?

The planes are perpendicular to each other

The planes are parallel to each other

Depends on the value of the vector

The planes are at an angle greater than 90°

If the planes \(A_1 x+B_1 y+C_1 z+D_1\)=0 and \(A_2 x+B_2 y+C_2 z+D_2\)=0 are at right angles to each other, then which of the following is true?

\(\frac{A_1+B_1+C_1}{A_2+B_2+C_2}\)=0

Find the angle between two planes \(\vec{r}.(2\hat{i}-\hat{j}+\hat{k})=3\) and \(\vec{r}.(3\hat{i}+2\hat{j}-3\hat{k})\)=5.

\(cos^{-1}⁡\frac{1}{\sqrt{22}}\)

\(cos^{-1}⁡\frac{1}{\sqrt{6}}\)

\(cos^{-1}⁡\frac{1}{\sqrt{132}}\)

\(cos^{-1}⁡\frac{1}{\sqrt{13}}\)

Which of the following is the correct formula for the angle between two planes \(A_1 x+B_1 y+C_1 z+D_1\)=0 and \(A_2 x+B_2 y+C_2 z+D_2\)=0?

cos⁡θ=\(\frac{A_1 B_1 C_1}{A_2 B_2 C_2}\)

cos⁡θ=\(\left |\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right |\)

sin⁡θ=\(\left |\frac{A_1 A_2-B_1 B_2-C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right |\)

cos⁡θ=\(A_1 A_2+B_1 B_2+C_1 C_2\)

Explore topic-wise MCQs in Mathematics Questions and Answers.

This section includes 5 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics Questions and Answers knowledge and support exam preparation. Choose a topic below to get started.

1.	Find the angle between the planes \(\vec{r}.(4\hat{i}+\hat{j}-2\hat{k})\)=6 and \(\vec{r}.(5\hat{i}-6\hat{j}+\hat{k})\)=7?
A.	\(cos^{-1}⁡\frac{12}{\sqrt{1302}}\)
B.	\(cos^{-1}⁡\frac{1}{\sqrt{1392}}\)
C.	\(cos^{-1}\frac{⁡23}{\sqrt{102}}\)
D.	\(cos^{-1}⁡\frac{15}{\sqrt{134}}\)
Answer» B. \(cos^{-1}⁡\frac{1}{\sqrt{1392}}\)

Discussion

2.	If two vectors \(\vec{r}.\vec{n_1}=d_1\) and \(\vec{r}.\vec{n_2}=d_2\) are such that \(\vec{n_1}.\vec{n_2}\)=0, then which of the following is true?
A.	The planes are perpendicular to each other
B.	The planes are parallel to each other
C.	Depends on the value of the vector
D.	The planes are at an angle greater than 90°
Answer» B. The planes are parallel to each other

Discussion

3.	If the planes \(A_1 x+B_1 y+C_1 z+D_1\)=0 and \(A_2 x+B_2 y+C_2 z+D_2\)=0 are at right angles to each other, then which of the following is true?
A.	\(\frac{A_1+B_1+C_1}{A_2+B_2+C_2}\)=0
B.	\(A_1+A_2+B_1 +B_2+C_1+C_2\)=0
C.	\(A_1+B_1+C_1=A_2 B_2 C_2\)
D.	\(A_1 A_2+B_1 B_2+C_1 C_2\)=0
Answer» E.

Discussion

4.	Find the angle between two planes \(\vec{r}.(2\hat{i}-\hat{j}+\hat{k})=3\) and \(\vec{r}.(3\hat{i}+2\hat{j}-3\hat{k})\)=5.
A.	\(cos^{-1}⁡\frac{1}{\sqrt{22}}\)
B.	\(cos^{-1}⁡\frac{1}{\sqrt{6}}\)
C.	\(cos^{-1}⁡\frac{1}{\sqrt{132}}\)
D.	\(cos^{-1}⁡\frac{1}{\sqrt{13}}\)
Answer» D. \(cos^{-1}⁡\frac{1}{\sqrt{13}}\)

Discussion

5.	Which of the following is the correct formula for the angle between two planes \(A_1 x+B_1 y+C_1 z+D_1\)=0 and \(A_2 x+B_2 y+C_2 z+D_2\)=0?
A.	cos⁡θ=\(\frac{A_1 B_1 C_1}{A_2 B_2 C_2}\)
B.	cos⁡θ=\(\left \|\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right \|\)
C.	sin⁡θ=\(\left \|\frac{A_1 A_2-B_1 B_2-C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right \|\)
D.	cos⁡θ=\(A_1 A_2+B_1 B_2+C_1 C_2\)
Answer» C. sin⁡θ=\(\left \|\frac{A_1 A_2-B_1 B_2-C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right \|\)

Discussion