Vector Algebra

Let \(\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}\) and \(\vec{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}\) be two vectors, such that \(\vec{a} \times \vec{b}=-\hat{i}+9 \hat{j}+12 \hat{k}\). Then the projection of \(\vec{b}-2 \vec{a}\) on $$\vec{b}...

Let \(\overrightarrow a\), \(\overrightarrow b\), \(\overrightarrow c\) be three non-coplanar vectors such that \(\overrightarrow a\) \(\times\) \(\overrightarrow b\) = 4\(\overrightarrow c\), \(\overrightarrow b\) \(\times\) $$\over...

If \(\overrightarrow a \,.\,\overrightarrow b = 1,\,\overrightarrow b \,.\,\overrightarrow c = 2\) and \(\overrightarrow c \,.\,\overrightarrow a = 3\), then the value of $$\left[ {\overrightarrow a \times \left( {\overrightarrow b \ti...

Let \(\overrightarrow{\mathrm{a}}=\alpha \hat{i}+\hat{j}-\hat{k}\) and \(\overrightarrow{\mathrm{b}}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0\). If the projection of \(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\) on th...

Let \(\theta\) be the angle between the vectors \(\overrightarrow a\) and \(\overrightarrow b\), where \(|\overrightarrow a | = 4,\) \(|\overrightarrow b | = 3\) and \(\theta \in \left( {{\pi \over 4},{\pi \over 3}} \right)\). Then $${...

Let \(\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k\) \({a_i} > 0\), \(i = 1,2,3\) be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of \(\overrightarrow a\) on...

Let \(\overrightarrow b = \widehat i + \widehat j + \lambda \widehat k\), \(\lambda\) \(\in\) R. If \(\overrightarrow a\) is a vector such that \(\overrightarrow a \times \overrightarrow b = 13\widehat i - \widehat j - 4\widehat k\) and...

Let \(\mathrm{ABC}\) be a triangle such that $$\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{CA}}=\overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{c}},|\overrightarrow{\ma...

Let \(\vec{a}=\hat{i}-\hat{j}+2 \hat{k}\) and let \(\vec{b}\) be a vector such that \(\vec{a} \times \vec{b}=2 \hat{i}-\hat{k}\) and \(\vec{a} \cdot \vec{b}=3\). Then the projection of \(\vec{b}\) on the vector \(\vec{a}-\vec{b}\) is :

Let \(\widehat a\), \(\widehat b\) be unit vectors. If \(\overrightarrow c\) be a vector such that the angle between \(\widehat a\) and \(\overrightarrow c\) is \({\pi \over {12}}\), and $$\widehat b = \overrightarrow c + 2\left( {\over...

Let \(\widehat a\) and \(\widehat b\) be two unit vectors such that \(|(\widehat a + \widehat b) + 2(\widehat a \times \widehat b)| = 2\). If \(\theta\) \(\in\) (0, \(\pi\)) is the angle between \(\widehat a\) and \(\widehat b\), then among...

Let \(\overrightarrow a\) and \(\overrightarrow b\) be two vectors such that \(\left| {2\overrightarrow a + 3\overrightarrow b } \right| = \left| {3\overrightarrow a + \overrightarrow b } \right|\) and the angle between $$\overrightarro...

Let \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) three vectors mutually perpendicular to each other and have same magnitude. If a vector \({ \overrightarrow r }\) satisfies.
$$\overrightarrow a \times \{ (\overrightarro...

Let \(\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b\) and \(\overrightarrow c = \widehat j - \widehat k\) be three vectors such that \(\overrightarrow a \times \overrightarrow b = \overrightarrow c\) and $...

Let \(\overrightarrow a = \widehat i + \widehat j + 2\widehat k\) and \(\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k\). Then the vector product $$\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\le...

Let \(\overrightarrow a = \widehat i - \alpha \widehat j + \beta \widehat k\),   \(\overrightarrow b = 3\widehat i + \beta \widehat j - \alpha \widehat k\) and \(\overrightarrow c = -\alpha \widehat i - 2\widehat j + \widehat k\), where ...

Let \(\overrightarrow a\), \(\overrightarrow b\) and \(\overrightarrow c\) be three vectors such that \(\overrightarrow a\) = \(\overrightarrow b\) \(\times\) (\(\overrightarrow b\) \(\times\) \(\overrightarrow c\)). If magnitudes of...

Let \(\overrightarrow a = \widehat i + 5\widehat j + \alpha \widehat k\), \(\overrightarrow b = \widehat i + 3\widehat j + \beta \widehat k\) and \(\overrightarrow c = - \widehat i + 2\widehat j - 3\widehat k\) be three vectors such tha...

If \(\overrightarrow a\) and \(\overrightarrow b\) are perpendicular, then $$\overrightarrow a \times \left( {\overrightarrow a \times \left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b } \right)} \rig...

If vectors \(\overrightarrow {{a_1}} = x\widehat i - \widehat j + \widehat k\) and \(\overrightarrow {{a_2}} = \widehat i + y\widehat j + z\widehat k\) are collinear, then a possible unit vector parallel to the vector $$x\widehat i + y\wi...

Let \(\overrightarrow a = \widehat i + \widehat j + \widehat k\) and \(\overrightarrow b = \widehat j - \widehat k\). If \(\overrightarrow c\) is a vector such that \(\overrightarrow a \times \overrightarrow c = \overrightarrow b\) an...

If the projection of the vector \(\widehat i + 2\widehat j + \widehat k\) on the sum of the two vectors \(2\widehat i + 4\widehat j - 5\widehat k\) and \(- \lambda \widehat i + 2\widehat j + 3\widehat k\) is 1, then \(\lambda\) is equal to...

A hall has a square floor of dimension 10 m \(\times\) 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is \({\cos ^{ - 1}}{1 \over 5}\), then the height of the hall (in meters) is :

Let \(\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k\) and \(\overrightarrow q = \widehat i + 2\widehat j + \widehat k\) be two vectors. If a vector $$\overrightarrow r = (\alpha \widehat i + \beta \widehat j + \gamma \wideha...

Let the vectors\((2 + a + b)\widehat i + (a + 2b + c)\widehat j - (b + c)\widehat k,(1 + b)\widehat i + 2b\widehat j - b\widehat k\) and \((2 + b)\widehat i + 2b\widehat j + (1 - b)\widehat k\), \(a,b,c, \in R\) be co-planar. Then which of ...

If \(\left( {\overrightarrow a + 3\overrightarrow b } \right)\) is perpendicular to \(\left( {7\overrightarrow a - 5\overrightarrow b } \right)\) and \(\left( {\overrightarrow a - 4\overrightarrow b } \right)\) is perpendicular to $$\lef...

If \(\left| {\overrightarrow a } \right| = 2,\left| {\overrightarrow b } \right| = 5\) and \(\left| {\overrightarrow a \times \overrightarrow b } \right|\) = 8, then \(\left| {\overrightarrow a .\,\overrightarrow b } \right|\) is equal to ...

Let a, b and c be distinct positive numbers. If the vectors \(a\widehat i + a\widehat j + c\widehat k,\widehat i+\widehat k\) and \(c\widehat i + c\widehat j + b\widehat k\) are co-planar, then c is equal to :

Let \(\overrightarrow a = \widehat i + 2\widehat j - \widehat k\), \(\overrightarrow b = \widehat i - \widehat j\) and \(\overrightarrow c = \widehat i - \widehat j - \widehat k\) be three given vectors. If \(\overrightarrow r\) is a ve...

Let \(\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k\) and \(\overrightarrow b = 3\widehat i - \alpha \widehat j + \widehat k\). If the area of the parallelogram whose adjacent sides are represented by the vectors $$\ove...

Let three vectors \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) be such that \(\overrightarrow c\) is coplanar with \(\overrightarrow a\) and \(\overrightarrow b\),
\(\overrightarrow a .\overrightarrow c\) ...

Let three vectors \(\overrightarrow a\), \(\overrightarrow b\) and \(\overrightarrow c\) be such that \(\overrightarrow a \times \overrightarrow b = \overrightarrow c\), $$\overrightarrow b \times \overrightarrow c = \overrightarrow...

Let a vector \({\overrightarrow a }\) be coplanar with vectors \(\overrightarrow b = 2\widehat i + \widehat j + \widehat k\) and \(\overrightarrow c = \widehat i - \widehat j + \widehat k\). If \({\overrightarrow a}\) is perpendicular to ...

If the shortest distance between the lines \(\overrightarrow {{r_1}} = \alpha \widehat i + 2\widehat j + 2\widehat k + \lambda (\widehat i - 2\widehat j + 2\widehat k)\), \(\lambda\) \(\in\) R, \(\alpha\) > 0 and $$\overrightarrow {{r_2}} ...

Let \(\overrightarrow a\), \(\overrightarrow b\), \(\overrightarrow c\) be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle \(\theta\), with the vector \(\overrightarrow a\) + $$\overrightarrow...

Let \(\overrightarrow a = 2\widehat i + \widehat j - 2\widehat k\) and \(\overrightarrow b = \widehat i + \widehat j\). If \(\overrightarrow c\) is a vector such that $$\overrightarrow a .\,\overrightarrow c = \left| {\overrightarrow c ...

For p > 0, a vector \({\overrightarrow v _2} = 2\widehat i + (p + 1)\widehat j\) is obtained by rotating the vector \({\overrightarrow v _1} = \sqrt 3 p\widehat i + \widehat j\) by an angle \(\theta\) about origin in counter clockwise direc...

In a triangle ABC, if \(\left| {\overrightarrow {BC} } \right| = 3\), \(\left| {\overrightarrow {CA} } \right| = 5\) and \(\left| {\overrightarrow {BA} } \right| = 7\), then the projection of the vector \(\overrightarrow {BA}\) on $$\overr...

Let \(\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k\) and \(\overrightarrow b = \widehat i + 2\widehat j - \widehat k\). Let a vector \(\overrightarrow v\) be in the plane containing \(\overrightarrow a\) and $$\overrightar...

A vector \(\overrightarrow a\) has components 3p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, $$\overr...

In a triangle ABC, if \(|\overrightarrow {BC} | = 8,|\overrightarrow {CA} | = 7,|\overrightarrow {AB} | = 10\), then the projection of the vector \(\overrightarrow {AB}\) on \(\overrightarrow {AC}\) is equal to :

Let \(\overrightarrow a\) and \(\overrightarrow b\) be two non-zero vectors perpendicular to each other and \(|\overrightarrow a | = |\overrightarrow b |\). If \(|\overrightarrow a \times \overrightarrow b | = |\overrightarrow a |\), the...

Let \(\overrightarrow a\) = 2\(\widehat i\) \(-\) 3\(\widehat j\) + 4\(\widehat k\) and \(\overrightarrow b\) = 7\(\widehat i\) + \(\widehat j\) \(-\) 6\(\widehat k\).If \(\overrightarrow r\) \(\times\) \(\overrightarrow a\) = $$\overri...

If \(\overrightarrow a = \alpha \widehat i + \beta \widehat j + 3\widehat k\),\(\overrightarrow b = - \beta \widehat i - \alpha \widehat j - \widehat k\) and \(\overrightarrow c = \widehat i - 2\widehat j - \widehat k\)such that $$\over...

Let \(\overrightarrow x\) be a vector in the plane containing vectors \(\overrightarrow a = 2\widehat i - \widehat j + \widehat k\) and \(\overrightarrow b = \widehat i + 2\widehat j - \widehat k\). If the vector \(\overrightarrow x\) i...

Let O be the origin. Let \(\overrightarrow {OP} = x\widehat i + y\widehat j - \widehat k\) and \(\overrightarrow {OQ} = - \widehat i + 2\widehat j + 3x\widehat k\), x, y\(\in\)R, x > 0, be such that $$\left| {\overrightarrow {PQ} } \righ...

Let a vector \(\alpha \widehat i + \beta \widehat j\) be obtained by rotating the vector \(\sqrt 3 \widehat i + \widehat j\) by an angle 45\(^\circ\) about the origin in counterclockwise direction in the first quadrant. Then the area of tri...

Let \(\overrightarrow a\) = \(\widehat i\) + 2\(\widehat j\) \(-\) 3\(\widehat k\) and \(\overrightarrow b = 2\widehat i\) \(-\) 3\(\widehat j\) + 5\(\widehat k\). If \(\overrightarrow r\) \(\times\) \(\overrightarrow a\) = $$\overright...

Let \(\overrightarrow c\) be a vector perpendicular to the vectors, \(\overrightarrow a\) = \(\widehat i\) + \(\widehat j\) \(-\) \(\widehat k\) and \(\overrightarrow b\) = \(\widehat i\) + 2\(\widehat j\) + \(\widehat k\). If $$\overrig...

If the vectors, \(\overrightarrow p = \left( {a + 1} \right)\widehat i + a\widehat j + a\widehat k\),

\(\overrightarrow q = a\widehat i + \left( {a + 1} \right)\widehat j + a\widehat k\) and

$$\overrightarrow r = a\widehat i + a\wideha...