By vector method, prove that cos(α + β) = cos α cos β -  sin α sin β

With usual notations, in any triangle ABC, prove by vector method that \(\frac { a }{ sinA } =\frac { b }{ sinB }=\frac { c }{ sinc }\)

Prove by vector method that sin(α −β) = sinα cosβ −cosα sinβ

If D is the midpoint of the side BC of a triangle ABC, then show by vector method that \({ \left| \vec { AB } \right| }^{ 2 }+{ \left| \vec { AC } \right| }^{ 2 }=2({ \left| \vec { AD} \right| }^{ 2 }+{ \left| \vec { BD } \right| }^{ 2 })\)

Prove by vector method that the perpendiculars (attitudes) from the vertices to the opposite sides of a triangle are concurrent.

In triangle, ABC the points, D, E, F are the midpoints of the sides BC, CA and AB respectively. Using vector method, show that the area of ΔDEF is equal to \(\frac{1}{4}\)(area of ΔABC )

If G is the centroid of a ΔABC, prove that (area of ΔGAB) = (area of ΔGBC) = (area of ΔGCA) = \(\frac{1}{3}\) (area of ΔABC)

Using vector method, prove that cos(α − β ) = cos α cos β +sin α sin β

Prove by vector method that sin(α + β ) = sin α cos β + cos α sin β

A particle acted on by constant forces \(8\hat { i } +2\hat { j } -6\hat { k } \) and \(6\hat { i } +2\hat { j } -2\hat { k } \) is displaced from the point (1, 2, 3) to the point (5, 4, 1). Find the total work done by the forces.

Forces of magnit \(5\sqrt { 2 } \) and \(10\sqrt { 2 } \) units acting in the directions \(\hat { 3i } +\hat { 4j } +\hat { 5k } \) and \(\hat { 10i } +\hat { 6j } -\hat { 8k } \) respectively, act on a particle which is displaced from the point with position vector \(\hat { 4i } -\hat { 3j } -\hat { 2k } \) to the point with position vector \(\hat { 6i } +\hat { j } -\hat { 3k } \). Find the work done by the forces.

Find the torque of the resultant of the three forces represented by \(-\hat { 3i } +\hat { 6j } +\hat { 3k } \), \(\hat { 4i } -\hat { 10j } +\hat { 12k } \) and \(\hat { 4i } +\hat { 7j } \)  acting at the point with position vector \(\hat { 8i } -\hat { 6j } -\hat { 4k } \), about the point with position vector \(\hat { 18i } +\hat { 3j } -\hat { 9k } \)

Show that the four points (6, -7, 0), (16, -19, -4), (0, 3, -6), (2, -5, 10) lie on a same plane.

If the vectors \(\vec { a } ,\vec { b } ,\vec { c } \) are coplanar, then prove that the vectors \(\vec { a } +\vec { b } ,\vec { b } +\vec { c } ,\vec { c } +\vec { a } \) are also coplanar.

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors represented by concurrent edges of a parallelepiped of volume 4 cubic units, find the value of \((\vec { a } +\vec { b } ).(\vec { b } \times \vec { c } )+(\vec { b } +\vec { c } ).(\vec { c } \times \vec { a } )+(\vec { c } +\vec { a } )(\vec { a } \times \vec { b } )\)

Find the altitude of a parallelepiped determined by the vectors \(\vec { a } =-2\hat { i } +5\hat { j } +3\hat { k } \), \(\hat { b } =\hat { i } +3\hat { j } -2\hat { k } \) and \(\vec { c } =-3\vec { i } +\vec { j } +4\vec { k } \) if the base is taken as the parallelogram determined by \(\vec { b } \) and \(\vec { c } \)

Determine whether the three vectors \(2\hat { i } +3\hat { j } +\hat { k } \), \(\hat { i } -2\hat { j } +2\hat { k } \) and \(\hat { 3i } +\hat { j } +3\hat { k } \) are coplanar.

Let \(\vec { a } =\hat { i } +\hat { j } +\hat { k } \),  \(\vec { b } =\hat { i } \)  and \(\vec { c } ={ c }_{ 1 }\hat { i } +{ c }_{ 2 }\hat { j } +{ c }_{ 3 }\hat { k } \). If \({ c }_{ 1 }=1\) and \({ c }_{ 2 }=2\), find \({ c }_{ 3 }\) such that \(\vec { a } ,\vec { b } \) and \(\vec { c } \) are coplanar.

If \(\vec { a } =\vec { i } -\vec { j } ,\vec { b } =\hat { i } -\hat { j } -4\hat { k } ,\vec { c } =3\hat { j } -\hat { k } \) and \(\vec { d } =2\hat { i } +5\hat { j } +\hat { k } \)
(i) \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )=[\vec { a } ,\vec { b } ,\vec { d } ]\vec { c } -[\vec { a } ,\vec { b } ,\vec { c } ]\vec { d } \)

If \(\vec { a } =2\hat { i } +3\hat { j } -\hat { k } ,\vec { b } =3\hat { i } +5\hat { j } +2\hat { k } ,\vec { c } =-\hat { i } -2\hat { j } +3\hat { k } \), verify that
(i) \((\vec { a } \times \vec { b } )\times \vec { c } =(\vec { a } .\vec { c } )\times \vec { b } -(\vec { b } .\vec { c } )\vec { a } \)
(ii) \(\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } .\vec { c } )\times \vec { b } -(\vec { a } .\vec { b } )\vec { c } \)

\(\vec { a } =2\hat { i } +3\hat { j } -\hat { k } ,\vec { b } =-\hat { i } +2\hat { j } -4\hat { k } ,\vec { c } =\hat { i } +\hat { j } +\hat { k } \) then find the value of \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )\).

If \(\vec { a } ,\vec { b } ,\vec { c } ,\vec { d } \) are coplanar vectors, then show that \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )=\vec { 0 } \).

If \(\vec { a } =\hat { i } +2\hat { j } +3\hat { k } ,\vec { b } =2\hat { i } -\hat { j } +\hat { k } ,\vec { c } =3\hat { i } +2\hat { j } +\hat { k } \) and \(\vec { a } \times (\vec { b } \times \vec { c } )\)= \(l\vec { a } +m\vec { b } +n\vec { c } \) , find the values of l, m, n.

A straight line passes through the point (1, 2, −3) and parallel to \(4\hat { i } +5\hat { j } -7\hat { k } \). Find 
(i) vector equation in parametric form
(ii) vector equation in non-parametric form
(iii) Cartesian equations of the straight line.

The vector equation in parametric form of a line is \(\vec { r } =(3\hat { i } -2\hat { j } +6\hat { k } )+t(2\hat { i } -\hat { j } +3\hat { k } )\). Find
(i) the direction cosines of the straight line
(ii) vector equation in non-parametric form of the line
(iii) Cartesian equations of the line.

Find the vector equation in parametric form and Cartesian equations of the line passing through (-4, 2, -3) and is parallel to the line  \(\frac { -x-2 }{ 4 } =\frac { y+3 }{ -2 } =\frac { 2z-6 }{ 3 } \)

Find the vector equation in parametric form and Cartesian equations of a straight passing through the points (-5, 7, 14) and (13, -5, 2). Find the point where the straight line crosses the xy - plane.

Find the non-parametric form of vector equation and Cartesian equations of the straight line passing through the point with position vector  \(4\hat { i } +3\hat { j } -7\hat { k } \) and parallel to the vector \(2\hat { i } -6\hat { j } +7\hat { k } \).

Find the parametric form of vector equation and Cartesian equations of the straight line passing through the point (−2, 3, 4) and parallel to the straight line \(\frac { x-1 }{ -4 } =\frac { y+3 }{ 5 } =\frac { 8-z }{ 6 } \)

Find the points where the straight line passes through (6,7, 4) and (8, 4,9) cuts the xz and yz planes.

Find the direction cosines of the straight line passing through the points (5, 6, 7) and (7, 9, 13). Also, find the parametric form of vector equation and Cartesian equations of the straight line passing through two given points.

Find the acute angle between the following lines
\(\vec { r } =(4\hat { i } -\hat { j } )+t(\hat { i } +2\hat { j } -2\hat { k } )\), \(\hat{r}=(\hat { i } +2\hat { j } -2\hat { k } )+s(\hat {- i } -2\hat { j } +2\hat { k } )\)

The vertices of ΔABC are A(7, 2, 1), B(6, 0, 3) , and C(4, 2, 4). Find ∠ABC .

If the straight line joining the points (2, 1, 4) and (a−1, 4, −1) is parallel to the line joining the points (0, 2, b −1) and (5, 3,  −2), find the values of a and b.

If the straight lines \(\frac { x-5 }{ 5m+2 } =\frac { 2-y }{ 5 } =\frac { 1-z }{ -1 } \) and \(x=\frac { 2y+1 }{ 4m } =\frac { 1-z }{ -3 } \) are perpendicular to each other, find the value of m.

Show that the points (2, 3, 4),(−1, 4, 5) and (8,1, 2) are collinear.

Find the point of intersection of the lines \(\frac { x-1 }{ 2 } =\frac { y-2 }{ 3 } =\frac { z-3 }{ 4 } \) and \(\frac { x-4 }{ 5 } =\frac { y-1 }{ 2 } =z\)

Find the parametric form of vector equation of a straight line passing through the point of intersection of the straight lines \(\vec { r } =(\hat { i } +\hat { 3j } -\hat { k } )+t(2\hat { i } +3\hat { j } +2\hat { k } )\) and \(\frac { x-2 }{ 1 } =\frac { y-4 }{ 2 } =\frac { z+3 }{ 4 } \) and perpendicular to both straight lines.

Determine whether the pair of straight lines \(\vec { r } (2\hat { i } +\hat { 6j } +\hat { 3k } )+t(2\hat { i } +3\hat { j } +4\hat { k } )\), \(\vec { r } =(2\hat { j } -3\hat { k } )+s(\hat { i } +2\hat { j } +3\hat { k } )\) are parallel. Find the shortest distance between them.

Find the coordinates of the foot of the perpendicular drawn from the point (-1, 2, 3) to the straight line \(\vec { r } =(\hat { i } -4\hat { j } +3\hat { k } )+t(2\hat { i } +3\hat { j } +\hat { k } )\). Also, find the shortest distance from the point to the straight line.

Find the parametric form of vector equation and Cartesian equations of a straight line passing through (5, 2,8) and is perpendicular to the straight lines 
\(\vec { r } =(\hat { i } +\hat { j } -\hat { k } )+s(2\hat { i } -2\hat { j } +\hat { k } )\)
\(\vec { r } =(\hat { 2i } -\hat { j } -3\hat { k } )+t(\hat { i } +2\hat { j } +2\hat { k } )\).

Show that the lines \(\vec { r } =(6\hat { i } +\hat { j } +2\hat { k } )+s(\hat { i } +2\hat { j } -3\hat { k } )\) and \(\vec { r } =(3\hat { i } +2\hat { j } -2\hat { k } )+t(2\hat { i } +4\hat { j } -5\hat { k } )\) are skew lines and hence find the shortest distance between them.

If the two lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ 3 } =\frac { z-1 }{ 4 } \) and \(\frac { x-3 }{ 1 } =\frac { y-m }{ 2 } =z\) intersect at a point, find the value of m

Show that the lines \(\frac { x-3 }{ 3 } =\frac { y-3 }{ -1 } =z-1=0\) and \(\frac { x-6 }{ 2 } =\frac { z-1 }{ 3 } ,y-2=0\) intersect. Also find the point of intersection.

Show that the straight lines x + 1=  2y = −12z and x = y + 2 = 6z − 6 are skew and hence find the shortest distance between them.

Find the parametric form of vector equation of the straight line passing through (−1, 2,1) and parallel to the straight line \(\vec { r } =(2\hat { i } +3\hat { j } -\hat { k } )+t(\hat { i } -2\hat { j } +\hat { k } )\) and hence find the shortest distance between the lines.

Find the foot of the perpendicular drawn from the point (5, 4, 2) to the line \(\frac { x+1 }{ 2 } =\frac { y-3 }{ 3 } =\frac { z-1 }{ -1 } \). Also, find the equation of the perpendicular.

Find the direction cosines of the normal to the plane 12x + 3y − 4z = 65. Also, find the non-parametric form of vector equation of a plane and the length of the perpendicular to the plane from the origin.

Find the vector and Cartesian equations of the plane passing through the point with position vector \(2\hat { i } +6\hat { j } +3\hat { k } \) and normal to the vector \(\hat { i } +3\hat { j } +5\hat { k } \)

A plane passes through the point (−1, 1, 2) and the normal to the plane of magnitude \(3\sqrt { 3 } \) makes equal acute angles with the coordinate axes. Find the equation of the plane.

If a plane meets the co-ordinate axes at A, B, C such that the centriod of the triangle ABC is the point (u, v, w), find the equation of the plane.

Find the vector parametric, vector non-parametric and Cartesian form of the equation of the plane passing through the points (-1, 2, 0), (2, 2, -1)and parallel to the straight line  \(\frac { x-1 }{ 1 } =\frac { 2y+1 }{ 2 } =\frac { z+1 }{ -1 } \)

Find the non-parametric form of vector equation, and Cartesian equation of the plane passing through the point (2, 3, 6) and parallel to the straight lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ 3 } =\frac { z-3 }{ 1 } \) and \(\frac { x+3 }{ 2 } =\frac { y-3 }{ -5 } =\frac { z+1 }{ -3 } \)

Find the non-parametric form of vector equation, and Cartesian equations of the plane passing through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9

Find parametric form of vector equation and Cartesian equations of the plane passing through the points (2, 2, 1), (1, −2, 3) and parallel to the straight line passing through the points (2, 1, −3) and (−1, 5, −8)

Find the non-parametric form of vector equation of the plane passing through the point (1, −2, 4) and perpendicular to the plane x + 2y −3z = 11 and parallel to the line \(\frac { x+7 }{ 3 } =\frac { y+3 }{ -1 } =\frac { z }{ 1 } \)

Find the parametric form of vector equation and Cartesian equations of the plane containing the line \(\vec { r } =(\hat { i } -\hat { j } +3\hat { k } )+t(2\hat { i } -\hat { j } +4\hat { k } )\) and perpendicular to plane \(\vec { r } .(\hat { i } +2\hat { j } +\hat { k } )=8\)

Find the parametric vector, non-parametric vector and Cartesian form of the equations of the plane passing through the points (3, 6, −2), (−1,−2, 6) , and (6, 4, −2).

Find the non-parametric form of vector equation, and Cartesian equations of the plane \(\vec { r } =(6\hat { i } -\hat { j } +\hat { k } )+s(-\hat { i } +2\hat { j } +\hat { k } )+(-5\hat { i } -4\hat { j } -5\hat { k } )\)

Show that the straight lines \(\vec { r } =(5\hat { i } +7\hat { j } -3\hat { k } )+s(-4\hat { i } +4\hat { j } -5\hat { k } )\) and \(\vec { r } =(8\hat { i } +4\hat { j } +5\hat { k } )+t(7\hat { i } +\hat { j } +3\hat { k } )\)are coplanar. Find the vector equation of the plane in which they lie.

Show that the lines \(\frac { x-2 }{ 1 } =\frac { y-3 }{ 1 } =\frac { z-4 }{ 3 } \) and \(\frac{x-1}{-3}=\frac{y-4}{2}=\frac{z-5}{1}\) coplanar. Also, find the plane containing these lines.

If the straight lines \(\frac { x-1 }{ 1 } =\frac { y-2 }{ 1 } =\frac { z-3 }{ { m }^{ 2 } } \) and \(\frac { x-3 }{ 1 } =\frac { y-2 }{ { m }^{ 2 } } =\frac { z-1 }{ 2 } \) are coplanar, find the distinct real values of m.

If the straight lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ 2 } \) and \(\frac { x-1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ \lambda } \) are coplanar, find λ and equations of the planes containing these two lines.

Show that the lines \(\vec { r } =(\hat {- i } -3\hat { j } -5\hat { k } )+s(3\hat { i } +5\hat { j } +7\hat { k } )\) and \(\vec { r } =(2\hat { i } +4\hat { j } +6\hat { k } )+t(\hat { i } +4\hat { j } +7\hat { k } )\) are coplanar. Also, find the non-parametric form of vector equation of the plane containing these lines

Find the equation of the plane passing through the line of intersection of the planes \(\vec { r } .(2\hat { i } -7\hat { j } +4\hat { k } )=3\) and 3x - 5y + 11 = 0, and the point (-2, 1, 3)

Find the equation of the plane passing through the line of intersection of the planes x + 2y + 3z = 2 and x - y + z = 3 and at a distance \(\frac { 2 }{ \sqrt { 3 } } \) from the point (3, 1, -1)

Find the equation of the plane which passes through the point (3, 4, -1) and is parallel to the plane 2x - 3y + 5z = 0. Also, find the distance between the two planes.

Find the length of the perpendicular from the point (1, -2, 3) to the plane x - y + z = 5.

Find the point of intersection of the line x - 1 = \(\frac { y }{ 2 } \) = z + 1 with the plane 2x - y + 2z = 2. Also, find the angle between the line and the plane.

Find the coordinates of the foot of the perpendicular and length of the perpendicular from the point ( 4, 3, 2) to the plane x + 2y + 3z = 2.

Find the acute angle between the following lines
\(\frac { x+4 }{ 3 } =\frac { y-7 }{ 4 } =\frac { z+5 }{ 5 } \), \(\vec { r } =4\hat { k } +t(2\hat { i } +\hat { j } +\hat { k } )\) 

Find the acute angle between the following lines
2x = 3y = −z and 6x = − y = −4z.