Angle between Two Vectors

Q. Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three vectors such that $\overrightarrow{a}=\overrightarrow{b}\times (\overrightarrow{b}\times \overrightarrow{c})$. If magnitudes of the vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are $\sqrt{2},1$ and $2$ respectively and the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\theta (0<\theta <\frac{\pi }{2})$, then the value of $1+\tan \theta$ is equal to

1. $2$
2. $\frac{\sqrt{3}+1}{\sqrt{3}}$
3. $1$
4. $\sqrt{3}+1$

Q.

Find the angle between two vectors a and b with magnitudes $\sqrt{3}$ and 2 respectively, having $a.b=\sqrt{6}$

Q.

The locus of a point that moves such that the difference of its distance from two fixed points is always a constant, is

1. a circle

2. a straight line

3. a hyperbola

4. an ellipse

Q.

$\overrightarrow{a}$ and $\overrightarrow{c}$ are unit vectors and $|\overrightarrow{b}|=4.$ The angle between $\overrightarrow{a}and\overrightarrow{c}isco{s}^{-1}\left(\frac{1}{4}\right).$
If $\overrightarrow{b}-2\overrightarrow{c}=\lambda \overrightarrow{a}$, then $\lambda$ is

1. $3or-4$

2. $\frac{1}{4}or\frac{3}{4}$

3. $-3or4$

4. $-\frac{1}{4}or\frac{3}{4}$

Q. Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors such that $|\overrightarrow{a}|=1,$ $|\overrightarrow{b}|=4$ and $\overrightarrow{a}\cdot \overrightarrow{b}=2.$ If $\overrightarrow{c}=(2\overrightarrow{a}\times \overrightarrow{b})-3\overrightarrow{b},$ then the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is

1. $\frac{\pi }{6}$
2. $\frac{\pi }{3}$
3. $\frac{5\pi }{6}$
4. $\frac{2\pi }{3}$

Q. If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ∠ABC. [∠ABC is the angle between the vectors and ]

Q. The rectangular component of a vector are (2, 2) .the corresponding rectangular component of another vector (1, 3) find the angle between two vectors

Q. If $p^{th},q^{th},r^{th}$ terms of a $G.P.$ are the positive numbers $a,b$ and $c,$ then the angle between the vectors $\left(\log a^2\right)\hat{i}+\left(\log b^2\right)\hat{j}+\left(\log c^2\right)\hat{k}$ and $(q-r)\hat{i}+(r-p)\hat{j}+(p-q)\hat{k}$ is

1. $\frac{\pi }{2}$
2. $\frac{\pi }{3}$
3. ${\sin }^{-1}\sqrt{a^2+b^2+c^2}$
4. ${\sin }^{-1}\sqrt{p^2+q^2+r^2}$

Q. The points O, A, B, C, D are such that $\overline{OA}=a,\overline{OB}=b,\overline{OC}=2a+3b$ and $\overline{OD}=a-2b$. If $|a|=3|b|$, then the angle between $\overline{BD}$ , $\overline{AC}$ is

1. $\frac{\pi }{3}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{6}$
4. $noneofthese$

Q.

Two vectors ${\overrightarrow{F}}_1$ and ${\overrightarrow{F}}_2$ each of magnitude are inclined to each other such that their resultant is to $\sqrt{3}F$ then the resultant of ${\overrightarrow{F}}_1$ and $-{\overrightarrow{F}}_2$ is

1. $F$

2. $\sqrt{2}F$

3. $\sqrt{3}F$

4. 2F

Q. Match the statements given in Column I with the values given in Column II

Column - I	Column - II
(A) If $\overrightarrow{a}=\hat{j}+\sqrt{3}\hat{k},\overrightarrow{b}=-\hat{j}+\sqrt{3}\hat{k}\text{and}\overrightarrow{c}=2\sqrt{3}\hat{k}$ form a triangle, then internal angle of the triangle between $\overrightarrow{a}\text{and}\overrightarrow{b}$ is	(p) $\frac{\pi }{6}$
(B) If ${\int }_a^b\left(f\right(x)-3x)dx=a^2-b^2,$ then the value of $f\left(\frac{\pi }{6}\right)$ is	(q) $\frac{2\pi }{3}$
(C) The value of $\frac{{\pi }^2}{\ln 3}{\int }_{\frac{7}{6}}^{\frac{5}{6}}\sec \left(\pi x\right)dx$ is	(r) $\frac{\pi }{3}$
(D) the maximum value of $\left|Arg\left(\frac{1}{1-z}\right)\right||z|=1,z\ne 1$ is given by	(s) $\pi$
	(t) $\frac{\pi }{2}$

2. A(r), B(p), C(s), D(t*)

Q. If $\overrightarrow{a}=4\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{b}=2\hat{i}-2\hat{j}+\hat{k}$, then find a unit vector parallel to the vector $\overrightarrow{a}+\overrightarrow{b}$

Q. If the vectors $\overrightarrow{a}=\left(clo{g}_{2}x\right)\hat{i}-6\hat{j}+2\hat{k}$ and $\overrightarrow{b}=\left(lo{g}_{2}x\right)\hat{i}+2\hat{j}+3\left(clo{g}_{2}x\right)\hat{k}$ make an obtuse angle for any $x\in (0,\infty )$ then c belongs to

1. $(-\infty ,-\frac{4}{3})$
2. $(-\infty ,0)$
3. $(-\frac{4}{3},0)$
4. $(-\frac{4}{3},\infty )$

Q.

If the sum of two unit vectors is a unit vector, then the magnitude of the difference is

1. $\sqrt{2}$

2. $\sqrt{3}$

3. $\frac{1}{\sqrt{2}}$

4. $\sqrt{5}$

Q.

The vector in the direction of the vector $\hat{i}-2\hat{j}+2\hat{k}$ that has magnitude 9 is

(a) $\hat{i}-2\hat{j}+2\hat{k}$

(b) $\frac{\hat{i}-2\hat{j}+2\hat{k}}{3}$

(c) $3(\hat{i}-2\hat{j}+2\hat{k})$

(d) $9(\hat{i}-2\hat{j}+2\hat{k})$

Q.

What is the value of ‘g’ when a body is thrown vertically upwards?

Q.

Does order matter for dot product?

Q. If $\overrightarrow{x}$ and $\overrightarrow{y}$ be two non-zero vectors such that $|\overrightarrow{x}+\overrightarrow{y}|=\left|\overrightarrow{x}\right|$ and $2\overrightarrow{x}+\lambda \overrightarrow{y}$ is perpendicular to $\overrightarrow{y}$, then the value of $\lambda$ is

Q.

If $\overrightarrow{a,}\overrightarrow{b},\overrightarrow{c}$ are three mutually perpendicular vectors equally inclined to $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$ at angle $\theta$ then find the value of $36{\cos }^{2}2\theta$.

Q.

Eliminate $\theta$ between $a=\cos$($\theta -\alpha$) and $b=\sin$($\theta -\beta$)

Q.

Choose the correct answer
If $\theta$ is the angle between two vectors a and b, then $a.b\ge 0$ only when
a) $0<\theta <\frac{\pi }{2}$
b) $0\le \theta <\frac{\pi }{2}$
c) $0<\theta <\pi$
d) $0\le \theta \le \pi$

Q. Let $\overrightarrow{u}$ be a vector coplanar with the vectors $\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{j}+\hat{k}$. If $\overrightarrow{u}$ is perpendicular to $\overrightarrow{a}$ and $\overrightarrow{u}\cdot \overrightarrow{b}=24,$ then $|\overrightarrow{u}{|}^2$ is equal to

1. $315$
2. $256$
3. $84$
4. $336$

Q. If $\overrightarrow{a},\overrightarrow{b}$ are unit vectors such that the vector $\overrightarrow{a}+3\overrightarrow{b}$ is perpendicular to $7\overrightarrow{a}-5\overrightarrow{b}$ and $\overrightarrow{a}-4\overrightarrow{b}$ is perpendicular to $7\overrightarrow{a}-2\overrightarrow{b}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

1. $\frac{\pi }{6}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{2}$

Q. The domain of the function f(x)=$\frac{\sqrt{-{\log }_{0.3}(x-1)}}{\sqrt{-x^2+2x+8}}$ is

1. $(1,4)$
2. none of these
3. $(2,4)$
4. $[2,4)$

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar vectors, then the number of real value(s) of $\lambda$ for which $\left[\lambda \right(\overrightarrow{a}+\overrightarrow{b}\left){\lambda }^2\overrightarrow{b}\lambda \overrightarrow{c}\right]=[\overrightarrow{a}\overrightarrow{b}+\overrightarrow{c}\overrightarrow{b}],$ is

Q. Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors such that $|\overrightarrow{a}|=1,$ $|\overrightarrow{b}|=4$ and $\overrightarrow{a}\cdot \overrightarrow{b}=2.$ If $\overrightarrow{c}$ is a vector such that $\overrightarrow{c}=(2\overrightarrow{a}\times \overrightarrow{b})-3\overrightarrow{b},$ then the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is

1. $\frac{\pi }{6}$
2. $\frac{\pi }{3}$
3. $\frac{5\pi }{6}$
4. $\frac{2\pi }{3}$

Q. If $\overrightarrow{\alpha }=a\hat{i}+b\hat{j}+c\hat{k},\overrightarrow{\beta }=b\hat{i}+c\hat{j}+a\hat{k}$ and $\overrightarrow{\gamma }=c\hat{i}+a\hat{j}+b\hat{k}$ be three coplanar vectors with $\overrightarrow{\alpha }\ne \overrightarrow{\beta }\ne \overrightarrow{\gamma }$ and $\overrightarrow{r}=\hat{i}+\hat{j}+\hat{k},$ then $\overrightarrow{r}$ is perpendicular to

1. $\overrightarrow{\alpha }$
2. $\overrightarrow{\beta }$
3. $\overrightarrow{\gamma }$
4. None of the above

Q. If the angle θ between the vectors $a=2x^2\hat{i}+4x\hat{j}+\hat{k}$ and $b=7\hat{i}-2\hat{j}+x\hat{k}$ is such that ${90}^{\circ }$ < $\theta$ < ${180}^{\circ }$
then x lies in the interval:

1. $(\frac{1}{2},1)$
2. $(0,\frac{1}{2})$
3. $(1,\frac{3}{2})$
4. $(\frac{1}{2},\frac{3}{2})$

Q. If $\overrightarrow{a},\overrightarrow{b}$ are unit vectors such that the vector $\overrightarrow{a}+3\overrightarrow{b}$ is perpendicular to $7\overrightarrow{a}-5\overrightarrow{b}$ and $\overrightarrow{a}-4\overrightarrow{b}$ is perpendicular to $7\overrightarrow{a}-2\overrightarrow{b}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

1. $\frac{\pi }{6}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{2}$

Q. Find direction cosines of vector $\hat{i}+2\hat{j}+3\hat{k}$.

1. $(\frac{1}{14},\frac{2}{14},\frac{3}{14})$
2. $(\sqrt{14},\frac{\sqrt{14}}{2},\frac{\sqrt{14}}{3})$
3. $(-\frac{1}{\sqrt{14}},-\frac{2}{\sqrt{14}},-\frac{3}{14})$
4. $(\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}})$