Vector Algebra

Let the position vectors of points 'A' and 'B' be i+j+k and 2i+j+3k, respectively. A point 'P' divides the line segment AB internally in the ratio λ:1(λ>0). If O is the origin and OB.OP - 3|OA*OP|² = 6, then λ is equal to 

Let a, b and c be three unit vectors such that |a-b|²+|a-c|²=8. Then |a+2b|²+|a+2c|² is equal to.

Let p = 2i + 3j + k and q = i + 2j + k be two vectors. If a vector r = (αi + βj + γk) is perpendicular to each of the vectors (p + q) and (p - q), and |r| = √3, then |α| + |β| + |γ| is equal to……

If f(x+y)=f(x)f(y) and Σₓ₌₁^∞ f(x)=2, x,y∈N where N is the set of all natural numbers, then the value of f(4)/f(2) is:

Let the vectors (2+a+b)i + (a+2b+c)j - (b+c)k, (1+b)i + 2bj - bk and (2+b)i + 2bj + (1-b)k, a,b,c ∈ R be co-planar. Then which of the following is true?

Let the volume of a parallelepiped whose coterminous edges are given by $\stackrel{}{}\stackrel{}{}\stackrel{}{}\stackrel{}{}\stackrel{}{}\stackrel{}{}\stackrel{}{}\stackrel{}{}$ $\stackrel{?}{u}=\hat{i}+\hat{j}+\lambda \hat{k},\stackrel{?}{v}=\hat{i}+\hat{j}+3\hat{k}$ and $\stackrel{?}{w}=2\hat{i}+\hat{j}+\hat{k}$ $\stackrel{}{}\stackrel{}{}\stackrel{}{}\stackrel{}{}$ be 1 cu. Unit. If $\theta$ $$ be the angle between the edges $\stackrel{?}{\mathrm{u}}$ $\stackrel{}{}$ and $\stackrel{}{}$ $\stackrel{?}{\mathrm{w}}$ , then $\mathrm{c}\mathrm{o}\mathrm{s}?\theta$ $$ can be:

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ . If $\lambda =\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}$ and $\vec{d}=\vec{a}*\vec{b}+\vec{b}*\vec{c}+\vec{c}*\vec{a}$ , then the ordered pair, $(\lambda ,\vec{d})$ is equal to: 

Three particles P, Q and R are moving along the vectors $\overrightarrow{A}=\widehat{i}+\widehat{j},\overrightarrow{B}=\widehat{j}+\widehat{k}$  and $\overrightarrow{C}=-\widehat{i}+\widehat{j}$  respectively. They strike on point and start to move in different directions. Now particle P is moving normal to the plane which contains vector $\overrightarrow{A}and\overrightarrow{B}.$  Similarly particle Q is moving normal to the plane which contains vector $\overrightarrow{A}and\overrightarrow{C}.$  The angle between the direction of motion of P and Q $co{s}^{-1}\left(\frac{1}{\sqrt[]{x}}\right).$  Then the value of x is_________.

Let $\overrightarrow{a}=\alpha \widehat{i}a+3\widehat{j}-\widehat{k},\overrightarrow{b}=3\widehat{i}-\beta \widehat{j}+4\widehat{k}and\overrightarrow{c}=\widehat{i}+2\widehat{j}-2\widehat{k}where\alpha ,\beta \in R,$ be three vectors. If the projection of $\overrightarrow{a}on\overrightarrow{c}is\frac{10}{3}and\overrightarrow{b}*\overrightarrow{c}=-6\widehat{i}+10\widehat{j}+7\widehat{k},$ then the value of + is equal to: