Vector Algebra ...

The vector $$b = 3j + 4k$$ is to be written as the sum of a vector $$b_{1}$$ parallel to $$a = i + j$$ and a vector $$b_{2}$$ perpendicular to $$a$$. Then $$b_{1}$$ is equal to

The magnitude of the scalar $$p$$ for which the vector $$p\left( -3\hat { i } -2\hat { j } +13\hat { k }  \right) $$ is of unit length is:

If $$\vec a, \vec b$$ and $$\vec c$$ are three non-coplanar vectors, then $$(\vec a +\vec  b -\vec  c) \cdot [(\vec a - \vec b) \times (\vec b - \vec c)$$ equals

Let $$\vec{v_1}, \vec{v_2}, \vec{v_3} , \vec{v_4} $$ be unit vectors in the xy - plane, one each in the interior of the four quadrants. Which of the following statements is necesserily true?

If the position vector $$\overrightarrow{a}$$ of the point $$(5, n)$$ is such that $$|\overrightarrow{a}|=13$$, then the value/values of n be

If $$\vec { a } ,\vec { b } ,\vec { c } $$ are mutually perpendicular unit vectors, then $$\left| \vec { a } +\vec { b } +\vec { c }  \right| $$ is equal to

Let $$u, v$$ and $$w$$ be such that $$\left| u \right| =1, \left| v \right| =3$$ and $$\left| w \right| =2$$. If the projection of $$v$$ along $$u$$ is equal to that of $$w$$ along $$u$$ and vectors $$v$$ and $$w$$ are perpendicular to each other, then $$\left| u-v+w \right| $$ equals

A vector $$R$$ is given by $$R=A\times \left( B\times C \right) $$, which of the following is true?

If $$a=\hat{i}+\hat{j}, b=2\hat{j}-\hat{k}$$ and $$r\times a=b\times a, r\times b=a\times b$$, then a unit vector in the direction of $${r}$$ is?

The resultant of $$P$$ and $$Q$$ is $$R$$. If $$Q$$ is doubled, $$R$$ is also doubled and if $$Q$$ is reversed, $$R$$ is again doubled. Then, $$P^{2} : Q^{2} : R^{2}$$ given by