Let v_1, v_2, v_3, v_4 be unit vectors in the | vedclass

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

Question

(C) Let the unit vectors be represented by their angles $	heta_1, 	heta_2, 	heta_3, 	heta_4$ where $	heta_1 \in (0, 90^{\circ})$,$	heta_2 \in (9

Vedclass · Accepted Answer

There exist $i, j$ with $1 \leq i < j \leq 4$ such that $v_i \cdot v_j < 0$

Vedclass · Answer

(C) Let the unit vectors be represented by their angles $	heta_1, 	heta_2, 	heta_3, 	heta_4$ where $	heta_1 \in (0, 90^{\circ})$,$	heta_2 \in (90^{\circ}, 180^{\circ})$,$	heta_3 \in (180^{\circ}, 270^{\circ})$,and $	heta_4 \in (270^{\circ}, 360^{\circ})$.$(a)$ The sum $v_1 + v_2 + v_3 + v_4$ is not necessarily zero. For example,if the vectors are very close to the positive $X$-axis and positive $Y$-axis,the sum will not be zero.$(b)$ The sum $v_i + v_j$ is not necessarily in the first quadrant. For instance,if $v_1$ is near $90^{\circ}$ and $v_2$ is near $180^{\circ}$,their sum will have a negative $X$-component.$(c)$ Consider $v_1$ in the first quadrant and $v_3$ in the third quadrant. The angle between them is $|	heta_1 - 	heta_3|$. Since $	heta_1 \in (0, 90^{\circ})$ and $	heta_3 \in (180^{\circ}, 270^{\circ})$,the difference $	heta_3 - 	heta_1$ lies in $(90^{\circ}, 270^{\circ})$. The dot product $v_1 \cdot v_3 = \cos(	heta_1 - 	heta_3)$. Since the angle difference can be greater than $90^{\circ}$,the cosine can be negative. Specifically,if we choose $v_1$ near $45^{\circ}$ and $v_3$ near $225^{\circ}$,the angle is $180^{\circ}$,and the dot product is $\cos(180^{\circ}) = -1 $(d)$ This is not necessarily true for all pairs,as shown in $(c)$.

Let $v_1, v_2, v_3, v_4$ be unit vectors in the $XY$-plane,one each in the interior of the four quadrants. Which of the following statements is necessarily true?

Similar Questions

If $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$,$\vec{b}=-\hat{i}+2\hat{j}+\hat{k}$,$\vec{c}=3\hat{i}+\hat{j}$ and $\vec{a}+\lambda\vec{b}$ is perpendicular to $\vec{c}$,then $\lambda=$

$A$ vector $a$ makes equal acute angles with the coordinate axes. Then the projection of vector $b = 5\hat{i} + 7\hat{j} - \hat{k}$ on $a$ is

In quadrilateral $ABCD$,$\overrightarrow{AB}=\vec{a}$,$\overrightarrow{BC}=\vec{b}$,$\overrightarrow{DA}=\vec{a}-\vec{b}$. $M$ is the midpoint of $BC$ and $X$ is a point on $DM$ such that $\overrightarrow{DX}=\frac{4}{5} \overrightarrow{DM}$. Then the points $A, X$ and $C$:

If $|a| + |b| = |c|$ and $a + b = c$,then the angle between $a$ and $b$ is

The value of $b$ such that the scalar product of the vector $(i + j + k)$ with the unit vector parallel to the sum of the vectors $(2i + 4j - 5k)$ and $(bi + 2j + 3k)$ is $1$,is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module