Let overline{a}, overline{b} and overline{c} | vedclass

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

Question

(A) Given,$|\overline{a}|=1, |\overline{b}|=1$ and $|\overline{c}|=2$. Using the vector triple product formula $\overline{a} 	imes (\overline{b} 	im

Vedclass · Accepted Answer

$\frac{\pi}{6}$

Vedclass · Answer

(A) Given,$|\overline{a}|=1, |\overline{b}|=1$ and $|\overline{c}|=2$. Using the vector triple product formula $\overline{a} 	imes (\overline{b} 	imes \overline{c}) = (\overline{a} \cdot \overline{c})\overline{b} - (\overline{a} \cdot \overline{b})\overline{c}$,we have: $\overline{a} 	imes (\overline{a} 	imes \overline{c}) = (\overline{a} \cdot \overline{c})\overline{a} - (\overline{a} \cdot \overline{a})\overline{c}$. Since $|\overline{a}|=1$,we have $\overline{a} \cdot \overline{a} = 1$. Substituting this into the given equation $\overline{a} 	imes (\overline{a} 	imes \overline{c}) + \overline{b} = \overline{0}$: $(\overline{a} \cdot \overline{c})\overline{a} - \overline{c} + \overline{b} = \overline{0}$. Rearranging gives $(\overline{a} \cdot \overline{c})\overline{a} - \overline{c} = -\overline{b}$. Taking the squared magnitude on both sides: $|(\overline{a} \cdot \overline{c})\overline{a} - \overline{c}|^2 = |-\overline{b}|^2$. $(\overline{a} \cdot \overline{c})^2 |\overline{a}|^2 + |\overline{c}|^2 - 2(\overline{a} \cdot \overline{c})(\overline{a} \cdot \overline{c}) = |\overline{b}|^2$. $(\overline{a} \cdot \overline{c})^2(1) + 4 - 2(\overline{a} \cdot \overline{c})^2 = 1$. $-(\overline{a} \cdot \overline{c})^2 = -3 \Rightarrow (\overline{a} \cdot \overline{c})^2 = 3$. Thus,$\overline{a} \cdot \overline{c} = \sqrt{3}$ (since the angle is acute,$\cos 	heta > 0$). $|\overline{a}||\overline{c}| \cos 	heta = \sqrt{3} \Rightarrow (1)(2) \cos 	heta = \sqrt{3}$. $\cos 	heta = \frac{\sqrt{3}}{2} \Rightarrow 	heta = \frac{\pi}{6}$.

Let $\overline{a}, \overline{b}$ and $\overline{c}$ be three vectors having magnitudes $1, 1$ and $2$ respectively. If $\overline{a} \times(\overline{a} \times \overline{c})+\overline{b}=\overline{0}$,then the acute angle between $\overline{a}$ and $\overline{c}$ is

Similar Questions

Wires $A$ and $B$ are made from the same material. $A$ has twice the diameter and three times the length of $B$. If the elastic limits are not reached,when each is stretched by the same tension,the ratio of energy stored in $A$ to that in $B$ is

Which of the following hormone levels causes the release of an ovum from the Graafian follicle (ovulation)?

Two uniform strings $A$ and $B$ made of steel are made to vibrate under the same tension. If the first overtone of $A$ is equal to the second overtone of $B$ and if the radius of $A$ is twice that of $B,$ the ratio of the lengths of the strings is

Which of the following statements is correct?

If $2x + 3y + 12 = 0$ and $x - y + 4\lambda = 0$ are conjugate with respect to the parabola $y^2 = 8x$,then $\lambda$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module