Consider the following statements :Statement | vedclass

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

Question

(A) Statement $(I)$ : The dot product of two vectors $\vec{a}$ and $\vec{b}$ is defined as $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos 	heta$.

Vedclass · Accepted Answer

Statement $(I)$ is false but Statement $(II)$ is false

Vedclass · Answer

(A) Statement $(I)$ : The dot product of two vectors $\vec{a}$ and $\vec{b}$ is defined as $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos 	heta$. If $|\vec{a}|=0$ or $|\vec{b}|=0$,then $\vec{a} \cdot \vec{b} = 0 	imes |\vec{b}| \cos 	heta = 0$ or $\vec{a} \cdot \vec{b} = |\vec{a}| 	imes 0 	imes \cos 	heta = 0$. Thus,Statement $(I)$ is true.Statement $(II)$ : The cross product of two vectors is defined as $\vec{a} 	imes \vec{b} = |\vec{a}| |\vec{b}| \sin 	heta \hat{n}$. If $\vec{a} 	imes \vec{b} = \vec{0}$,then $\sin 	heta = 0$,which implies $	heta = 0$ or $	heta = \pi$. This means the vectors are parallel or collinear,not perpendicular. Thus,Statement $(II)$ is false.

Consider the following statements :
Statement $(I)$ : If either $|\vec{a}|=0$ or $|\vec{b}|=0$,then $\vec{a} \cdot \vec{b}=0$.
Statement $(II)$ : If $\vec{a} \times \vec{b}=\vec{0}$,then $\vec{a}$ is perpendicular to $\vec{b}$.
Which of the following is correct?

Similar Questions

Let $OA = a, OB = b$ be two non-collinear vectors,$OP = x_1 a + y_1 b, OQ = x_2 a + y_2 b$ and $A^{\prime}O = OA, B^{\prime}O = OB$. If $x_1 = -\frac{3}{4}, x_2 = \frac{1}{3}, y_1 = \frac{7}{4}, y_2 = \frac{5}{3}$,then

If $a = \hat{i} + 2 \hat{j} + 3 \hat{k}$,$b = 2 \hat{i} + 3 \hat{j} + \hat{k}$,$c = 8 \hat{i} + 13 \hat{j} + 9 \hat{k}$ and $x a + y b + z c = 0$,then $\frac{x y}{z^2} =$

Let $p = (x + 4y)\vec{a} + (2x + y + 1)\vec{b}$ and $q = (y - 2x + 2)\vec{a} + (2x - 3y - 1)\vec{b}$,where $\vec{a}$ and $\vec{b}$ are non-collinear vectors. If $3p = 2q$,then the values of $x$ and $y$ are:

If $a$ and $b$ are the position vectors of $A$ and $B$ respectively,then the position vector of a point $C$ on $AB$ produced such that $\overrightarrow{AC} = 3\overrightarrow{AB}$ is

If $a = 2i + j - 8k$ and $b = i + 3j - 4k,$ then the magnitude of $a + b = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Consider the following statements :Statement $(I)$ : If either $|\vec{a}|=0$ or $|\vec{b}|=0$,then $\vec{a} \cdot \vec{b}=0$.Statement $(II)$ : If $\vec{a} \times \vec{b}=\vec{0}$,then $\vec{a}$ is perpendicular to $\vec{b}$.Which of the following is correct?