Collinear Vectors

Q.

If $a=i-j,b=j-k,c=k-i.$ If $d$ is a unit vector such that $a.d=0=[bcd],$ then $d$ is (are) equal to $?$

1. $\pm \frac{(i+j-k)}{\sqrt{3}}$

2. $\pm \frac{(i+j-2k)}{\sqrt{6}}$

3. $\pm \frac{(i+j+k)}{\sqrt{3}}$

4. $\pm k$

5. $i+j$

Q.

Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three non -zero vectors such that no two of these are collinear. If the vector $\overrightarrow{a}+2\overrightarrow{b}$ is collinear with $\overrightarrow{c}$ and $\overrightarrow{b}+3\overrightarrow{c}$ is collinear with $\overrightarrow{a}$ $(\lambda$ being some non -zero scalar) then $\overrightarrow{a}+2\overrightarrow{b}+6\overrightarrow{c}$ equals

1. $\lambda \overrightarrow{a}$
   

2. $\lambda \overrightarrow{b}$
   

3. $\lambda \overrightarrow{c}$
   

4. $0$

Q.

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three non-zero vectors, no two of which are collinear, $\overrightarrow{a}+\overrightarrow{b}$ is collinear with $\overrightarrow{c}$ and $\overrightarrow{b}+3\overrightarrow{c}$ is collinear with $\overrightarrow{a}$, then $|\overrightarrow{a}+2\overrightarrow{b}+6\overrightarrow{c}|$ will be equal to

1. 9

2. 1

3. None of these

4. zero

Q. The three points whose position vectors are $\hat{i}+2\hat{j}+3\hat{k},3\hat{i}+4\hat{j}+7\hat{k}and-3\hat{i}-2\hat{j}-5\hat{k}$

1. form the vertices of an equilateral triangle
2. form the vertices of a right angled triangle
3. are collinear
4. form the vertices of an isosceles triangle.

Q. Let a, b and c be three nonzero vectors, no two of which are collinear. If the vector a+2b is collinear with c, and b+3c is collinear with a, then a+2b+6c =

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

Q. If $\overrightarrow{a}$ is a vector of magnitude 50 and is collinear with the vector $\overrightarrow{b}=6\hat{i}-8\hat{j}-\frac{15}{2}\hat{k}$ and makes an obtuse angle with the positive direction of z - axis, then $\overrightarrow{a}$ is equal to

1. $24\hat{i}-32\hat{j}-30\hat{k}$
2. $24\hat{i}+32\hat{j}+30\hat{k}$
3. $12\hat{i}-16\hat{j}-15\hat{k}$
4. $Noneofthese$

Q.

If the points A, B and C have position vectors (2, 1, 1), (6, -1, 2) and (14, -5, P) respectively and if they are collinear, then P =

1. 2

2. 8

3. 4

4. 6

Q. If $\overrightarrow{a}\times \overrightarrow{b}$ is defined as $\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|$ $\sin \theta$ where $\theta$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ and it is given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are collinear vectors, then $\overrightarrow{a}\times \overrightarrow{b}$ =;___

Q.

The points with position vectors $60\hat{i}+3\hat{j},40\hat{i}-8\hat{j},a\hat{i}-52\hat{j}$ are collinear, if

1. a = -40

2. a = 40

3. None of these

4. a = 20