A particle acted upon by constant forces \(\hat { 2j } +\hat { 5j } +\hat { 6k } \) and \(-\hat { i } -\hat { 2j } -\hat { k } \)  is displaced from the point (4, −3, −2) to the point (6, 1, −3). Find the total work done by the forces.

Find the magnitude and the direction cosines of the torque about the point (2, 0, -1) of a force \((\hat { 2i } +\hat { j } -\hat { k } )\), whose line of action passes through the origin

Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is \(\frac { 1 }{ 2 } \left| \vec { AC } \times \vec { BD } \right| \).

Find the magnitude and direction cosines of the torque of a force represented by \(\hat { 3i } +\hat { 4j } -\hat { 5k } \) about the point with position vector \(\hat { 2i } -\hat { 3j } +\hat { 4k } \) acting through a point whose position vector is \(\hat { 4i } +\hat { 2j } -\hat { 3k } \).

Find the volume of the parallelepiped whose coterminus edges are given by the vectors \(\hat { 2i } -\hat { 3j } +\hat { 4k } \), \(\hat { i } +\hat { 2j } -\hat { k } \) and \(\hat {3 i } -\hat { j } +\hat { 2k } \)

Show that the vectors \(\hat { i } +\hat { 2j } -\hat { 3k } \), \(\hat { 2i } -\hat { j } +\hat { 2k } \) and \(\hat { 3i } +\hat { j } -\hat { k } \)

If \(\hat { 2i } -\hat { j } +\hat { 3k } ,\hat { 3i } +\hat { 2j } +\hat { k } ,\hat { i } +\hat { mj } +\hat { 4k } \) are coplanar, find the value of m.

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three vectors, prove that \([\vec { a } +\vec { c } ,\vec { a } +\vec { b } ,\vec { a } +\vec { b } +\vec { c } ]\) = \([\vec { a } ,\vec { b } ,\vec { c } ]\)

The volume of the parallelepiped whose coterminus edges are \(7\hat { i } +\lambda \hat { j } -3\hat { k } ,\hat { i } +2\hat { j } -\hat { k } \), \(-3\hat { i } +7\hat { j } +5\hat { k } \) is 90 cubic units. Find the value of λ.

Prove that \([\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } ]\) = \([{ \vec { a } ,\vec { b } ,\vec { c } }]^{ 2 }\)

Prove that \((\vec { a } .(\vec { b } \times \vec { c } ))\vec { a } =(\vec { a } \times \vec { b } )\times (\vec { a } \times \vec { c } )\)

If \(\vec { a } =-2\hat { i } +3\hat { j } -2\hat { k } ,\vec { b } =3\hat { i } -\hat { j } +3\hat { k } ,\vec { c } =2\hat { i } -5\hat { j } +\hat { k } \) find \((\vec { a } \times \vec { b } )\times \vec { c } \) and \((\vec { a } \times \vec { b } )\times \vec { c } \). State whether they are equal.

If \(\vec { a } =\hat { i } -2\hat { j } +3\hat { k } ,\vec { b } =2\hat { i } +\hat { j } -2\hat { k } ,\vec { c } =3\hat { i } +2\hat { j } +\hat { k } \) find 
(i) \((\vec { a } \times \vec { b } )\times \vec { c } \)
(ii) \(\vec { a } \times (\vec { b } \times \vec { c } )\)

Find the angle between the straight line \(\frac { x+3 }{ 2 } =\frac { y-1 }{ 2 } =-z\) with coordinate axes.

Find the angle between the lines \(\vec { r } =(\hat { i } +2\hat { j } +4\hat { k } )+t(2\hat { i } +2\hat { j } +\hat { k } )\) and the straight line passing through the points (5, 1, 4) and (9, 2, 12)

Find the angle between the straight lines \(\frac { x-4 }{ 2 } =\frac { y }{ 1 } =\frac { z-1 }{ -2 } \) and \(\frac { x-4 }{ 2 } =\frac { y }{ 1 } =\frac { z-1 }{ -2 } \) and state whether they are parallel or perpendicular.

Show that the straight line passing through the points A (6, 7, 5) and B(8, 10, 6) is perpendicular to the straight line passing through the points C(10, 2, -5) and D(8, 3, -4) 

Show that the lines \(\frac { x-1 }{ 4 } =\frac { 2-y }{ 6 } =\frac { z-4 }{ 12 } \) and \(\frac { x-3 }{ -2 } =\frac { y-3 }{ 3 } =\frac { 5-z }{ 6 } \) are parallel.

Find the shortest distance between the two given straight lines \(\vec { r } =(2\hat { i } +3\hat { j } +4\hat { k } )+t(-2\hat { i } +\hat { j } -2\hat { k } )\) and \(\frac { x-3 }{ 2 } =\frac { y }{ -1 } =\frac { z+2 }{ 2 } \)

Find the direction cosines of the normal to the plane and length of the perpendicular from the origin to the plane \(\vec { r } .(3\hat { i } -4\hat { j } +12\hat { k } )=5\)

Find the vector equation of a plane which is at a distance of 7 units from the origin having 3,−4, 5 as direction ratios of a normal to it.

Find the intercepts cut off by the plane \(\vec { r } .(6\hat { i } +4\hat { j } -3\hat { k } )\) = 12 on the coordinate axes.

Find the non-parametric form of vector equation, and Cartesian equation of the plane passing through the point (0, 1, -5) and parallel to the straight lines \(\vec { r } =(\hat { i } +2\hat { j } -4\hat { k } )+s(\hat { i } +3\hat { j } +6\hat { k } )\) and \(\hat { r } =(\hat { i } -3\hat { j } +5\hat { k } )+t(\hat { i } +\hat { j } -\hat { k } )\)

Find the angle between the straight line \(\vec { r } =(2\hat { i } +\hat { j } +\hat { k } )+t(\hat { i } -\hat { j } +\hat { k } )\) and the plane 2x-y+z = 5

Find the acute angle between the planes \(\vec { r } .(2\hat { i } +2\hat { j } +2\hat { k } )\) and 4x-2y+2z = 15.

Find the distance of a point (2, 5, −3) from the plane \(\vec { r } .(6\hat { i } -3\hat { j } +2\hat { k } )\) = 5

Find the distance of the point (5, -5, -10) from the point of intersection of a straight line passing through the points A (4, 1, 2) and B (7, 5, 4) with the plane x - y + z = 5

Find the distance between the parallel planes x + 2y - 2z + 1 = 0 and 2x + 4y - 4z + 5 = 0

Find the distance between the planes \(\vec { r } .(2\hat { i } -\hat { j } -2\hat { k } )\) = 6 and \(\vec { r } .(6\hat { i } -\hat { 3j } -\hat { 6k } )\) = 27

Find the equation of the plane passing through the intersection of the planes \(\vec { r } .(\hat { i } +\hat { j } +\hat { k } )+1=0\) and \(\vec { r } .(2\hat { i } -3\hat { j } +5\hat { k } )=2\) and the point (-1, 2, 1).

Find the equation of the plane passing through the intersection of the planes 2x + 3y −z + 7 = 0 and and x +y −2z + 5 = 0 and is perpendicular to the plane x +y −3z −5 = 0.

Find the image of the point whose position vector is \(\hat { i } +2\hat { j } +3\hat { k } \) in the plane \(\vec { r } .(\hat { i } +2\hat { j } +4\hat { k } )\) = 38

Find the coordinates of the point where the straight line \(\vec { r } =(2\hat { i } -\hat { j } +2\hat { k } )+t(3\hat { i } +4\hat { j } +2\hat { k } )\) intersects the plane x−y+z−5 = 0.

If \(\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}, \vec{b}=b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k} \text { and } \vec{c}=c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}\) then \((\vec{a} \times \vec{b}) \cdot \vec{c}=\left|\begin{array}{ccc} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array}\right|\)

For any three vectors \(\vec{a}, \vec{b}, \text { and } \vec{c},(\vec{a} \times \vec{b}) \cdot \vec{c}=\vec{a} \cdot(\vec{b} \times \vec{c})\)

The scalar triple product preserves addition and scalar multiplication. That is
\( {[(\vec{a}+\vec{b}), \vec{c}, \vec{d}] } =[\vec{a}, \vec{c}, \vec{d}]+[\vec{b}, \vec{c}, \vec{d}] \)
\({[\lambda \vec{a}, \vec{b}, \vec{c}] } =\lambda[\vec{a}, \vec{b}, \vec{c}], \forall \lambda \in \mathbb{R} \)
\({[\vec{a},(\vec{b}+\vec{c}), \vec{d}] } =[\vec{a}, \vec{b}, \vec{d}]+[\vec{a}, \vec{c}, \vec{d}] \)
\({[\vec{a}, \lambda \vec{b}, \vec{c}] } =\lambda[\vec{a}, \vec{b}, \vec{c}], \forall \lambda \in \mathbb{R} \)
\({[\vec{a}, \vec{b},(\vec{c}+\vec{d})] } =[\vec{a}, \vec{b}, \vec{c}]+[\vec{a}, \vec{b}, \vec{d}] \)
\({[\vec{a}, \vec{b}, \lambda \vec{c}] } =\lambda[\vec{a}, \vec{b}, \vec{c}], \forall \lambda \in \mathbb{R}\)

The scalar triple product of three non-zero vectors is zero if, and only if, the three vectors are coplanar.

The scalar triple product of three non-zero vectors is zero if, and only if, the three vectors are coplanar.

Three vectors \(\vec{a}, \vec{b}, \vec{c}\) are coplanar if, and only if, there exist scalars r, s,t \(\in \mathbb{R}\) such that atleast one of them is non-zero and \(r \vec{a}+s \vec{b}+t \vec{c}=\overrightarrow{0}\)

If \(\vec{a}, \vec{b}, \vec{c} \text { and } \vec{p}, \vec{q}, \vec{r}\) are any two systems of three vectors, and if \(\vec{p}=x_{1} \vec{a}+y_{1} \vec{b}+z_{1} \vec{c}\) \(\vec{q}=x_{2} \vec{a}+y_{2} \vec{b}+z_{2} \vec{c}, \text { and, } \vec{r}=x_{3} \vec{a}+y_{3} \vec{b}+z_{3} \vec{c}\) then \([\vec{p}, \vec{q}, \vec{r}]=\left|\begin{array}{lll} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{array}\right|[\vec{a}, \vec{b}, \vec{c}]\)

Find the vector equation of the plane passing through the point (2, 2, 3) having 3, 4, 3 as direction ratios of the normal to the plane.

For any three vectors \(\vec{a}, \vec{b}, \vec{c}\) we have \(\vec{a} \times(\vec{b} \times \vec{c})=(\vec{a} \cdot \vec{c}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{c}\)

For any three vectors \(\vec{a}, \vec{b}, \vec{c}\) we have \(\vec{a} \times(\vec{b} \times \vec{c})+\vec{b} \times(\vec{c} \times \vec{a})+\vec{c} \times(\vec{a} \times \vec{b})=\overrightarrow{0}\) . (Jacobi’s identity)

For any four vectors \(\vec{a}, \vec{b}, \vec{c}, \vec{d}\) we have \((\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})=\left|\begin{array}{ll} \vec{a} \cdot \vec{c} & \vec{a} \cdot \vec{d} \\ \vec{b} \cdot \vec{c} & \vec{b} \cdot \vec{d} \end{array}\right|\) . (Lagrange’s identity)

The vector equation of a straight line passing through a fixed point with position vector \(\vec{a} \) parallel to a given vector \(\vec{b} \text { is } \vec{r}=\vec{a}+t \vec{b} \text {, where } t \in \mathbb{R}\)

The parametric form of vector equation of a line passing through two given points whose position vectors are \(\vec{a} \text { and } \vec{b}\) respectively is \(\vec{r}=\vec{a}+t(\vec{b}-\vec{a}), t \in \mathbb{R}\)

The shortest distance between the two parallel lines \(\vec{r}=\vec{a}+s \vec{b} \text { and } \vec{r}=\vec{c}+t \vec{b}\) is given by \(d=\frac{|(\vec{c}-\vec{a}) \times \vec{b}|}{|\vec{b}|} \text {, where }|\vec{b}| \neq 0 \text {. }\) 

The shortest distance between the two skew lines \(\vec{r}=\vec{a}+s \vec{b} \text { and } \vec{r}=\vec{c}+t \vec{d}\) is given by \(\delta=\frac{|(\vec{c}-\vec{a}) \cdot(\vec{b} \times \vec{d})|}{|\vec{b} \times \vec{d}|}, \text { where }|\vec{b} \times \vec{d}| \neq 0\)

The equation of the plane at a distance p from the origin and perpendicular to the unit normal vector \(\hat{d} \text { is } \vec{r} \cdot \hat{d}=p .\)

The general equation ax + by + cz + d = 0 of first degree in x, y, z represents a plane.

If three non-collinear points with position vectors \(\vec{a}, \vec{b}, \vec{c}\) are given, then the vector equation of the plane passing through the given points in parametric form is 
\(\vec{r}=\vec{a}+s(\vec{b}-\vec{a})+t(\vec{c}-\vec{a}), \text { where } \vec{b} \neq 0, \vec{c} \neq 0 \text { and } s, t \in \mathbb{R}\) 

The acute angle \(\theta\) between the planes \(a_{1} x+b_{1} y+c_{1} z+d_{1}=0\) and \(a_{2} x+b_{2} y+c_{2} z+ d_{2}=0\) is given by \(\theta=\cos ^{-1}\left(\frac{\left|a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}\right|}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\right)\)

The acute angle \(\theta\) between the two planes \(\vec{r} \cdot \vec{n}_{1}=p_{1} \text { and } \vec{r} \cdot \vec{n}_{2}=p_{2}\) is given by \(\theta=\cos ^{-1}\left(\frac{\left|\vec{n}_{1} \cdot \vec{n}_{2}\right|}{\left|\vec{n}_{1}\right|\left|\vec{n}_{2}\right|}\right)\)

The perpendicular distance from a point with position vector \(\vec{u}\) to the plane \(\vec{r} \cdot \vec{n}=p\) is given by \(\delta=\frac{|\vec{u} \cdot \vec{n}-p|}{|\vec{n}|}\) 

The distance between two parallel planes \(a x+b y+c z+d_{1}=0 \text { and } a x+b y+c z+d_{2}=0\) is given by \(\frac{\left|d_{1}-d_{2}\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}\)

The vector equation of a plane which passes through the line of intersection of the planes \(\vec{r} \cdot \vec{n}_{1}=d_{1} \text { and } \vec{r} \cdot \vec{n}_{2}=d_{2}\) is given by \(\left(\vec{r} \cdot \vec{n}_{1}-d_{1}\right)+\lambda\left(\vec{r} \cdot \vec{n}_{2}-d_{2}\right)=0\) where \(\lambda \in \mathbb{R}\) .

The position vector of the point of intersection of the straight line \(\vec{r}=\vec{a}+t \vec{b}\) and the plane \(\vec{r} \cdot \vec{n}=p \text { is } \vec{a}+\left(\frac{p-(\vec{a} \cdot \vec{n})}{\vec{b} \cdot \vec{n}}\right) \vec{b}, \text { provided } \vec{b} \cdot \vec{n} \neq 0\)