The angle between two diagonals of a cube is:

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the vertices of the cube be $O(0,0,0)$,$A(a,0,0)$,$B(a,a,0)$,$C(a,a,a)$,$D(0,a,a)$,$E(0,0,a)$,$F(a,0,a)$,and $G(0,a,0)$.Consider two diagonals

Vedclass · Accepted Answer

$\cos ^{-1}\left(\frac{1}{3}\right)$

Vedclass · Answer

(A) Let the vertices of the cube be $O(0,0,0)$,$A(a,0,0)$,$B(a,a,0)$,$C(a,a,a)$,$D(0,a,a)$,$E(0,0,a)$,$F(a,0,a)$,and $G(0,a,0)$.Consider two diagonals of the cube,for example,the diagonal connecting $(0,0,0)$ to $(a,a,a)$ and the diagonal connecting $(a,0,0)$ to $(0,a,a)$.The vector along the first diagonal is $\vec{v_1} = a\hat{i} + a\hat{j} + a\hat{k}$.The vector along the second diagonal is $\vec{v_2} = -a\hat{i} + a\hat{j} + a\hat{k}$.The angle $	heta$ between these two vectors is given by $\cos 	heta = \frac{|\vec{v_1} \cdot \vec{v_2}|}{|\vec{v_1}| |\vec{v_2}|}$.$\vec{v_1} \cdot \vec{v_2} = (a)(-a) + (a)(a) + (a)(a) = -a^2 + a^2 + a^2 = a^2$.$|\vec{v_1}| = \sqrt{a^2 + a^2 + a^2} = a\sqrt{3}$.$|\vec{v_2}| = \sqrt{(-a)^2 + a^2 + a^2} = a\sqrt{3}$.$\cos 	heta = \frac{a^2}{(a\sqrt{3})(a\sqrt{3})} = \frac{a^2}{3a^2} = \frac{1}{3}$.Therefore,$	heta = \cos ^{-1}\left(\frac{1}{3}ight)$.

Similar Questions

For what values of $x$ is the angle between the vectors $\vec{a} = x\hat{i} - 3\hat{j} - \hat{k}$ and $\vec{b} = 2x\hat{i} + x\hat{j} - \hat{k}$ acute,and the angle between the vector $\vec{b}$ and the $y$-axis obtuse?

Find the projection of the vector $\hat{i}+3 \hat{j}+7 \hat{k}$ on the vector $7 \hat{i}-\hat{j}+8 \hat{k}$.

What is the projection vector of the vector $\vec{a} = (1, 1, 1)$ onto the vector $\vec{b} = (2, 2, 1)$?

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=\hat{i}-\hat{j}+\hat{k}$,and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ are three vectors,then the vector $\vec{r}$ in the plane of $\vec{a}$ and $\vec{b}$,whose projection on $\vec{c}$ is $\frac{1}{\sqrt{3}}$,is given by:

Let $A=(3,4,0), B=(4,4,4), C=(-6,2,3)$ and $D=(1,1,2)$. If $\theta$ is the acute angle between the lines $AB$ and $CD$,then $\cos \theta=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module