Vector Algebra ...

The angle between the two vectors $$\hat { i } +\hat { j } +\hat { k }$$ and $$ 2\hat { i } -2\hat { j } +2\hat { k } $$ is equal to

Let $$u, v$$ and $$w$$ be vectors such that $$u + v + w = 0$$. If $$|u| = 3, |v| = 4$$ and $$|w| = 5$$, then $$u . v + v . w + w.u$$ is equal to

If $$a=\hat { i } +\hat { j } +\hat { k } $$, $$b=4\hat { i } +3\hat { j } +4\hat { k }$$ and $$c=\hat { i } +\alpha \hat { j } +\beta \hat { k } $$ are coplanar and $$\left| c \right| =\sqrt { 3 } $$, then

If $$\vec{a} = \hat{i} + 2 \hat{j} + 2 \hat{k} , |\vec{b}| = 5 $$ and the angle between $$\vec{a}$$ and $$\vec{b}$$ is $$ \dfrac{\pi}{6} $$, then the area of the triangle formed by these two vectors as two sides is 

If $$\lambda (3 \widehat i + 2 \widehat j - 6 \widehat k)$$ is a unit vector, then the values of $$\lambda $$ are

Let $$a,b,c$$ be three non-zero vectors such that $$a+b+c=0$$, then $$\lambda b\times a+b\times c+c\times a=0$$, where $$\lambda$$ is

Let $$ABCD$$ be a parallelogram. If $$AB=\hat { i } +3\hat { j } +7\hat { k } , AD=2\hat { i } +3\hat { j } -5\hat { k } $$ and $$p$$ is a unit vector parallel to $$AC$$, then $$p$$ is equal to

If $$a=2i+2j+3k, b=-1+2j+k$$ and $$c = 3i + j$$, then $$a+tb$$ this perpendicular to $$c$$; if $$t$$ is equal to 

Let $$P\left( 1,2,3 \right) $$ and $$Q\left( -1,-2,-3 \right) $$ be the two points and let $$O$$ be the origin. Then, $$\left| PQ+OP \right| $$ is equal to

If $$3p + 2q =i + j + k$$ and $$3p - 2q = i - j - k$$, then the angle between $$p$$ and $$q$$ is