Test for Collinearity of Vectors

Q.

How do you prove vectors are collinear $?$

Q.

If $a^{1/x}=b^{1/y}=c^{1/z}$ and $a,b,c$are in geometric progression, then $x,y,z$ are in

1. AP

2. GP

3. HP

4. None of these

Q.

Find a vector of magnitude 5 units and parallel to the resultant of the vectors $a=2\hat{i}+3\hat{j}-\hat{k}andb=\hat{i}-2\hat{j}+\hat{k}.$

Q.

If vectors, $\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k}$, $\overrightarrow{b}=\hat{i}+2\hat{j}-3\hat{k}$, $\overrightarrow{c}=3\hat{i}+\lambda \hat{j}+5\hat{k}$ are coplanar. Then the value of $\lambda$ is equal to?

1. $\lambda =4$

2. $\lambda =-4$

3. $\lambda =-7$

4. $\lambda =7$

Q.

Let $\stackrel{\rightharpoonup }{a}$, $\stackrel{\rightharpoonup }{b}$ and $\stackrel{\rightharpoonup }{c}$be three non - zero vectors that are pairwise non-collinear. If $\stackrel{\rightharpoonup }{a}+3\stackrel{\rightharpoonup }{b}$ is collinear with $\stackrel{\rightharpoonup }{c}$ and $\stackrel{\rightharpoonup }{b}+2\stackrel{\rightharpoonup }{c}$ is collinear with $\stackrel{\rightharpoonup }{a}$, then $\stackrel{\rightharpoonup }{a}+3\stackrel{\rightharpoonup }{b}+6\stackrel{\rightharpoonup }{c}$ is equal to ?

1. $0$

2. $a$

3. $a+b$

4. $c$

Q.

Is the cross product always perpendicular?

Q. Let $\overrightarrow{\alpha }=(\lambda -2)\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{\beta }=(4\lambda -2)\overrightarrow{a}+3\overrightarrow{b}$ be two given vectors where vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-collinear. The value of $\lambda$ for which vectors $\overrightarrow{\alpha }$ and $\overrightarrow{\beta }$ are collinear, is:

1. $3$
2. $4$
3. $-4$
4. $-3$

Q. Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be non-zero vectors such that no two are collinear and $(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}=\left(\frac{1}{3}\right)|\overrightarrow{b}||\overrightarrow{c}|\overrightarrow{a}$. If $\theta$ is the angle between the vectors $\overrightarrow{b}$ and $\overrightarrow{c}$, then the value of sin $\theta$ is

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

Q. Find the value of $x$ such that the points $A(3,2,1),B(4,x,5),C(4,2,-2)\text{and}D(6,5,-1)$ are coplanar.

Q.

If $\overrightarrow{a}=2\hat{i}-2\hat{j}+3\hat{k},\overrightarrow{b}=3\hat{i}-2\hat{j}+4\hat{k}$ are two vectors then the dot product of vectors$a$ and $b$ will be

Q.

What is the sum of cross-products?

Q. Let $\overrightarrow{u}$ be a vector on rectangular coordinate system with sloping angle ${60}^{\circ }.$ Suupose that $|\overrightarrow{u}-\hat{i}|$ is geometric mean of $|\overrightarrow{u}|$ and $|\overrightarrow{u}-2\hat{i}|,$ then the value of $2(\sqrt{2}+1)|\overrightarrow{u}|=$

Q. The vectors $2\overrightarrow{i}-3\overrightarrow{j}+4\overrightarrow{k},\overrightarrow{i}-2\overrightarrow{j}+3\overrightarrow{k}$ and $3\overrightarrow{i}+\overrightarrow{j}-2\overrightarrow{k}$

1. are linearly dependent
2. are linearly independent
3. are coplanar
4. form an isosceles triangle

Q. Find the value of $p$ for which the vectors $3\hat{i}+2\hat{j}+9\hat{k}\text{and}\hat{i}-2p\hat{j}+3\hat{k}$ are parallel.

Q.

Show that $|a|b+|b|a$ is perpendicular to $|a|b-|b|a$ for any two non-zero vectors a and b.

Q. Three vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are such that $\overrightarrow{a}\times \overrightarrow{b}=4(\overrightarrow{a}\times \overrightarrow{c})$ and $|\overrightarrow{a}|=|\overrightarrow{b}|=1$ and $|\overrightarrow{c}|=\frac{1}{4}$. If the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{\pi }{3}$, then $\overrightarrow{b}$ is

1. $\overrightarrow{a}+4\overrightarrow{c}$
2. $\overrightarrow{a}-4\overrightarrow{c}$
3. $4\overrightarrow{c}-\overrightarrow{a}$
4. $2\overrightarrow{c}-\overrightarrow{a}$

Q. Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non-zero and non-collinear vectors. Then which of the following is/are always CORRECT?

1. $\overrightarrow{a}\times \overrightarrow{b}=\left[\overrightarrow{a}\overrightarrow{b}\hat{i}\right]\hat{i}+\left[\overrightarrow{a}\overrightarrow{b}\hat{j}\right]\hat{j}+\left[\overrightarrow{a}\overrightarrow{b}\hat{k}\right]\hat{k}$
2. $\overrightarrow{a}\cdot \overrightarrow{b}=(\overrightarrow{a}\cdot \hat{i})(\overrightarrow{b}\cdot \hat{i})+(\overrightarrow{a}\cdot \hat{j})(\overrightarrow{b}\cdot \hat{j})+(\overrightarrow{a}\cdot \hat{k})(\overrightarrow{b}\cdot \hat{k})$
3. If $\overrightarrow{u}=\hat{a}-(\hat{a}\cdot \hat{b})\hat{b}$ and $\overrightarrow{v}=\hat{a}\times \hat{b},$ then $\left|\overrightarrow{u}\right|=\left|\overrightarrow{v}\right|$
4. If $\overrightarrow{c}=\overrightarrow{a}\times (\overrightarrow{a}\times \overrightarrow{b})$ and $\overrightarrow{d}=\overrightarrow{b}\times (\overrightarrow{a}\times \overrightarrow{b}),$ then $\overrightarrow{c}+\overrightarrow{d}=\overrightarrow{0}$

Q. Let$P\equiv (\alpha ,\beta )$ be a point on the line $3x+2y+10=0$ such that $|PA-PB|$ is maximum and $A\equiv (4,2)$ , $B\equiv (2,4)$, then the value of $|\alpha |+|\beta |$ is equal to

Q. Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three unit vectors such that no two of the vectors are collinear. If the vector $\overrightarrow{a}+\overrightarrow{b}$ is collinear with $\overrightarrow{c}$ and the vector $\overrightarrow{b}+\overrightarrow{c}$ is collinear with $\overrightarrow{a},$ then the value of $|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|$ is

Q.

What is the physical meaning of cross product?

Q. The points whose position vectors are $60i+3j,40i-8j$ and $ai-52j$ collinear, if

1. $a=-40$
2. $a=40$
3. $a=20$
4. $a=-20$

Q.

Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ , $\overrightarrow{b}=\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{c}=x\hat{i}+(x-2)\hat{j}-\hat{k}$ are three vectors. If vector $\overrightarrow{c}$ lies in the plane of $\overrightarrow{a}$ and $\overrightarrow{b}$, then $x$ equals to:

1. $0$

2. $1$

3. $-4$

4. $-2$

Q. Suppose that $\overrightarrow{p},\overrightarrow{q}$ and $\overrightarrow{r}$ are three non-coplanar vectors in $R^3.$ Let the components of a vector $\overrightarrow{s}$ along $\overrightarrow{p},\overrightarrow{q},$ and $\overrightarrow{r}$ be $4,3,$ and $5,$ respectively. If the components of this vector $\overrightarrow{s}$ along $(-\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r}),$ $(\overrightarrow{p}-\overrightarrow{q}+\overrightarrow{r}),$ and $(-\overrightarrow{p}-\overrightarrow{q}+\overrightarrow{r}),$ are $x,y,$ and $z,$ respectively, then the value of $2x+y+z$ is

Q. Let $\overline{a},\overline{b},\overline{c}$ be three non-zero vectors, no two of which are collinear. lf the vector $\overline{a}+2\overline{b}$ is collinear with $\overline{c}$ and $\overline{b}+3\overline{c}$ is collinear with $\overline{a}$ , then $\overline{a}+\overline{b}+3\overline{c}=$

1. $\lambda \overline{b}$
2. $\lambda \overline{a}$
3. $\lambda \overline{c}$
4. $0$

Q. The area of the parallelogram whose diagonals are $\hat{i}-3\hat{j}+2\hat{k},-\hat{i}+2\hat{j}$ is

1. $4\sqrt{29}sq$.units
2. $10\sqrt{3}sq$.units
3. $\frac{1}{2}\sqrt{21}sq$.units
4. $\frac{1}{2}\sqrt{270}sq$.units

Q. If the vectors $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ $(a\ne b\ne c\ne 1)$ are coplanar , then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is equal to

1. $0$
2. $2$
3. $1$
4. none of these

Q. For any vector $\overrightarrow{x}$, the value of $(\overrightarrow{x}\times \hat{i}{)}^2+(\overrightarrow{x}\times \hat{j}{)}^2+(\overrightarrow{x}\times \hat{k}{)}^2$ is equal to

1. $|\overrightarrow{x}{|}^2$
2. $2|\overrightarrow{x}{|}^2$
3. $3|\overrightarrow{x}{|}^2$
4. $4|\overrightarrow{x}{|}^2$

Q. If $a>0$ and $A,B,C$ are variable angles of $△ABC,$ such that $a\tan A+2a\tan B+3a\tan C=7a,$ then the minimum integral value of ${\tan }^{2}A+{\tan }^{2}B+{\tan }^{2}C=$

Q. If $\overrightarrow{a}=-\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}\text{and}\overrightarrow{b}=2\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k},$ then the vector $\overrightarrow{c}$ satisfying the conditions
$1)\text{coplanar with}\overrightarrow{a}\text{and}\overrightarrow{b}$
$2)\text{perpendicular to}\overrightarrow{b}$
$3)\overrightarrow{a}\cdot \overrightarrow{c}=7$ is

1. $-\frac{3}{2}\hat{i}+\frac{5}{2}\hat{j}+3\hat{k}$
2. $-\frac{7}{3}\hat{i}+\frac{7}{3}\hat{j}+\frac{7}{3}\hat{k}$
3. $-6\hat{i}+\hat{k}$
4. $-\hat{k}$

Q. The points with position vectors $\overrightarrow{a}+\overrightarrow{b},\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{a}+k\overrightarrow{b}$ are collinear for all real values of $k$.