Bisector of Angle between Two Vectors

Q.

Find the value of $\lambda$ such that the vectors $\overrightarrow{a}=2\hat{i}+\lambda \hat{j}+\hat{k}and\overrightarrow{b}=\hat{i}+2\hat{j}+3\hat{k}$ are orthogonal.

(a) 0

(b) 1

(c) $\frac{3}{2}$

(d) $\frac{-5}{2}$

Q. Let $\overrightarrow{a}=\hat{i}+5\hat{j}+\alpha \hat{k},\overrightarrow{b}=\hat{i}+3\hat{j}+\beta \hat{k}$ and $\overrightarrow{c}=-\hat{i}+2\hat{j}-3\hat{k}$ be three vectors such that, $|\overrightarrow{b}\times \overrightarrow{c}|=5\sqrt{3}$ and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$. Then the greatest amongst the values of $|\overrightarrow{a}{|}^2$ is

Q. The vector c directed along the internal bisector of the angle between the vectors $a=7\hat{i}-4\hat{j}-4\hat{k}andb=-2\hat{i}-\hat{j}+2\hat{k}$ with $|c|=5\sqrt{6}$, is

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

Q. { If }\overline a and }\overline b are non-collinear vectors, then the value of }x for which vectors }\overlineα=(x-2)\overline a+\overline b an }}{\overlineβ=(3+2x)\overline a-2\overline b are collinear, is given by

Q. Prove that if modulus A vector is equal to modulus B vector but vector A is not parallel to vector B then vector A-B is perpendicular to vector A+B

Q. The position vectors of two points A and C are $9\hat{i}-\hat{j}+7\hat{k}$ and $7\hat{i}-2\hat{j}+7\hat{k}$ respectively. The point of intersection of the lines containing vectors $\overrightarrow{AB}=4\hat{i}-\hat{j}+3\hat{k}\text{and}\overrightarrow{CD}=2\hat{i}-\hat{j}+2\hat{k}$ is P. If a vector $\overrightarrow{PQ}$ is perpendicular to $\overrightarrow{AB}$~\text {and }\overrightarrow{CD} \text { and }PQ = 15\) units, the possible position vectors of Q are $x_1\hat{i}+x_2\hat{j}+x_3\hat{k}\text{and}{y}_1\hat{i}+y_2\hat{j}+y_3\hat{k}$. Then the value of $\sum _{i=1}^3(x_i+y_i)$ is equal to

Q. The vector $a\hat{i}+b\hat{j}+c\hat{k}$ is a bisector of the angle between the vectors $\hat{i}+\hat{j}$ and $\hat{j}+\hat{k}$ if

1. a=b
2. a=c
3. c=a+b
4. a =b=c

Q. { The vectors }3\vec a-5\vec b and }2\vec a+\vec b are mutually }}{ perpendicular and the vectors }\vec a+4\vec b and }}{-\vec a+\vec b are also mutually perpendicular. Then the }}{ angle between the vectors }\vec a and }\vec b, is

Q.

The value of $\lambda$ for which the vectors $3\hat{i}-6\hat{j}+\hat{k}and2\hat{i}-4\hat{j}+\lambda \hat{k}$ are parallel, is

(a) $\frac{2}{3}$ (b) $\frac{3}{2}$ (c) $\frac{5}{2}$ (d) $\frac{2}{5}$

Q. Let $\overrightarrow{c}$ be a vector perpendicular to the vectors $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}+\hat{k}.$ If $\overrightarrow{c}\cdot (\hat{i}+\hat{j}+3\hat{k}),$ then the value of $\overrightarrow{c}\cdot (\overrightarrow{a}\times \overrightarrow{b})$ is equal to

Q.

The vector c directed along the internal bisector of the angle between the vectors a = 7i - 4j - 4k and b = -2i - j + 2k with |c| = 5√6 ?

Q. The vector c directed along the internal bisector of the angle between the vectors $a=7\hat{i}-4\hat{j}-4\hat{k}$ and $b=-2\hat{i}-\hat{j}+2\hat{k}$ with $|c|=5\sqrt{6}$, is

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

Q. Let $\overrightarrow{\alpha }=\hat{i}+\hat{j}+\hat{k},\overrightarrow{\beta }=\hat{i}-\hat{j}-\hat{k}$ and $\overrightarrow{\gamma }=-\hat{i}+\hat{j}-\hat{k}$ be three vectors. A vector $\overrightarrow{\delta },$ in the plane of $\overrightarrow{\alpha }$ and $\overrightarrow{\beta },$ whose projection on $\overrightarrow{\gamma }$ is $\frac{1}{\sqrt{3}},$ is given by

1. $-\hat{i}-3\hat{j}-3\hat{k}$
2. $\hat{i}-3\hat{j}-3\hat{k}$
3. $-\hat{i}+3\hat{j}+3\hat{k}$
4. $\hat{i}+3\hat{j}-3\hat{k}$

Q. The vector which joins the point $A(4,5,6)$ to $B(10,11,12)$ is

1. $2\hat{i}+2\hat{j}+2\hat{k}$
2. $4\hat{i}+4\hat{j}+4\hat{k}$
3. $6\hat{i}+6\hat{j}+6\hat{k}$
4. $8\hat{i}+8\hat{j}+8\hat{k}$

Q. A vector $\overrightarrow{r}$ is inclined at equal angles to the three axes. If the magnitude of $\overrightarrow{r}$ is $2\sqrt{3}$, find $\overrightarrow{r}$. [NCERT EXEMPLAR]

Q.

What if the cross product is $\overrightarrow{0}$?

Q. The vector c directed along the internal bisector of the angle between the vectors $a=7\hat{i}-4\hat{j}-4\hat{k}andb=-2\hat{i}-\hat{j}+2\hat{k}$ with $|c|=5\sqrt{6}$, is

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

Q. If $4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$ and $2\hat{i}+5\hat{j}+7\hat{k}$ are the position vectors of the vertices $A,B$ and $C$, respectively of triangle $ABC$, then the position vector of the point where the bisector of $\angle A$ meets $BC$ is

1. $\frac{2}{3}(-6\hat{i}-8\hat{j}-6\hat{k})$
2. $\frac{2}{3}(6\hat{i}+8\hat{j}+6\hat{k})$
3. $\frac{1}{3}(6\hat{i}+13\hat{j}+18\hat{k})$
4. $\frac{1}{3}(5\hat{j}+12\hat{k})$

Q. Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $\hat{i}+2\hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively, in the ratio $2:1$,
(i) internally
(ii) externally

Q. Find the area bounded by the curve $y=\sqrt{x},x=2y+3$ in the first quadrant and x-axis.

Q. If $4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$ and $2\hat{i}+5\hat{j}+7\hat{k}$ are the position vectors of the vertices $A,B$ and $C$, respectively of triangle $ABC$, then the position vector of the point where the bisector of $\angle A$ meets $BC$ is

1. $\frac{2}{3}(-6\hat{i}-8\hat{j}-6\hat{k})$
2. $\frac{2}{3}(6\hat{i}+8\hat{j}+6\hat{k})$
3. $\frac{1}{3}(6\hat{i}+13\hat{j}+18\hat{k})$
4. $\frac{1}{3}(5\hat{j}+12\hat{k})$

Q. If $4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$ and $2\hat{i}+5\hat{j}+7\hat{k}$ are the position vectors of the vertices A, B and C respectively of triangle ABC. The position vector of the point where the bisector of angle A meets BC is

1. $\frac{2}{3}(-6\hat{i}-8\hat{j}-6\hat{k})$
2. $\frac{2}{3}(6\hat{i}+8\hat{j}+6\hat{k})$
3. $\frac{1}{3}(6\hat{i}+13\hat{j}+18\hat{k})$
4. $\frac{1}{3}(5\hat{i}+12\hat{k})$

Q. The cartesian equation of a line is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$ write its vector form.

Q. Let $P,Q,R$ and $S$ be the points on the plane with position vectors $-2\hat{i}-\hat{j},4\hat{i},3\hat{i}+3\hat{j}$ and $-3\hat{i}+2\hat{j}$respectively. The quadrilateral $PQRS$ must be a

1. Parallelogram, which is neither a rhombus nor a rectangle
2. Rectangle, but not a square
3. Square
4. Rhombus, but not a square

Q. The angle between the two diagonals of a cube is
(a) 30°

(b) 45°

(c) ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

(d) ${\cos }^{-1}\left(\frac{1}{3}\right)$

Q. Find the vector which acts like the angle bisector for the vectors $2\hat{i}+2\hat{j}+\hat{k}and2\hat{i}+4\hat{j}+\sqrt{5}\hat{k}$

1. 
2. 
3. 
4. 

Q. Show that four points whose position vectors are $6\hat{i}-7\hat{j},16\hat{i}-19\hat{j}-4\hat{k},3\hat{i}-6\hat{k},2\hat{i}-5\hat{j}+10\hat{k}$ are coplanar.

Q. If $(x,y,z)\ne (0,0,0)$ and $(\hat{i}+\hat{j}+3\hat{k})x+(3\hat{i}-3\hat{j}+\hat{k})y+(-4\hat{i}+5\hat{j})z=\lambda (x\hat{i}+y\hat{j}+z\hat{k})$, then the value of $\lambda$ will be

1. $-1,0$
2. $0,-2$
3. $-2,0$
4. $0,-1$

Q. The vector $\overrightarrow{C}$, directed along the internal bisector of the angle between the vectors $\overrightarrow{a}=7\hat{i}-4\hat{j}-4\hat{k}$ and $\overrightarrow{b}=-2\hat{i}-\hat{j}+2\hat{k}$ with $|\overrightarrow{c}|=5\sqrt{6}$, is

1. $\frac{5}{3}(5\hat{i}-5\hat{j}+2\hat{k})$
   
2. $\pm \left(\frac{5}{3}\right(\hat{i}-7\hat{j}+2\hat{k})$
3. $\frac{5}{3}(\hat{i}-7\hat{j}+2\hat{k})$
   
4. $\frac{5}{3}(-5\hat{i}-5\hat{j}+2\hat{k})$

Q. If $4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$ and $2\hat{i}+5\hat{j}+7\hat{k}$ are the position vectors of the vertices A, B and C respectively of triangle ABC. The position vector of the point where the bisector of angle A meets BC is

1. $\frac{2}{3}(-6\hat{i}-8\hat{j}-6\hat{k})$
2. $\frac{1}{3}(5\hat{i}+12\hat{k})$
3. $\frac{2}{3}(6\hat{i}+8\hat{j}+6\hat{k})$
4. $\frac{1}{3}(6\hat{i}+13\hat{j}+18\hat{k})$