If a lines makes angle 60° and 45° with the positive directions of x-axis and z-axis respectively, then find the angle that it makes with the y-axis.

2

(a)

If a line makes angle \(\alpha ,\beta ,\gamma \) with the coordinates axis ,then find the value of cos \(cos2\alpha +cos2\beta +cos2\gamma \)

2

(a)

Let \(I_{ e }m_{ i }n_{ i }i=1,2,3\) be the direction cosines of three mutually perpendicular vector ion space
\( \left[ \begin{matrix} { l }_{ 4 } & { m }_{ 1 } & { n }_{ 1 } \\ { l }_{ 2 } & { m }_{ 2 } & { n }_{ 2 } \\ { l }_{ 3 } & { m }_{ 3 } & { n }_{ 3 } \end{matrix} \right] \)
Show that AA^'= l₃, where A = 

2

(a)

If the equation of a line \(\frac { x-2 }{ 2 } =\frac { 2y-5 }{ -3 } ,z=-1\) then find the ratio of the line and a point on the line.

2

(a)

If the equation of a line is x = ay + b z = cy + d, then find direction ratios of the line and a point on the line.

2

(a)

Find the distance between the point (5, ,4, - 6) and its image in xy-plane.

2

(a)

Find the angle between the lines
\(\overset { \rightarrow }{ r } =(2i-5j+k)+l(i+j+3k)\) and \(\overset { \rightarrow }{ r } =(i-j-k)+\mu (2i-3j+k)\)

2

(a)

If three consecutive vertices of a parallelogram are (3, - 1, - 1), (5, - 4,0), (2,3, - 2), then the coordinates of the fourth vertex .

2

(a)

Show that the line through the points (0, 3, 2), (3, 5, 6) is perpendicular of the line through the points (1, - 1, 2) and (3,4, - 2).

2

(a)

Find vector equation as well as cartesian equation which is passes through the point (1, 2, 3) and is parallel to the vector 3i + 2j - 3k.

2

(a)

Find the cartesian and vector equation of the line which passes through the point (- 2, 4, - 5) and parallel to the line given by \(\frac { x+3 }{ 3 } =\frac { y-4 }{ 5 } =\frac { x-z }{ -6 } \)

2

(a)

Find the vector equation of the plane which is at a distance of 5 units from the origin and normal to the plane is \(3\check { i } -2\check { j } +6\check { k } \)

2

(a)

Find the angle between the planes 7x + 2y + 6z = 15 and 3x-y + 10z = 17.

2

(a)

Find the angle between line \(\frac { x-1 }{ 6 } =\frac { y+3 }{ 2 } =\frac { z-2 }{ 3 } \) and the plane 2x - Y + 2z - 13 = 0.

2

(a)

A plane meets the co-ordinate axes in A, B, and C such that the centroid of \(\triangle\)ABC is the point \((\alpha,\beta,\gamma).\) Show that the equation of the plane is \({{x}\over{\alpha}}+{{y}\over{\beta}}+{{z}\over{\gamma}}=3.\)

2

(a)