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Find the direction cosines of a line segment whose direction ratios are:

(i) 2. – 6, 3

(ii) 2, - 1, - 2,

(iii) – 9, 6, -2

Find the direction ratios and the direction cosines of the line segment joining the points:

(i) A (1, 0, 0) and B(0, 1, 1)

(ii) A(5, 6, -3) and B (1, -6, 3)

(iii) A (-5, 7, -9) and B (-3, 4, -6)

Show that the line joining the points A(1,-1, 2) and B(3, 4, -2) is perpendicular to the line joining the points C(0, 3, 2) and D (3, 5, 6).

Show that the line segment joining the origin to the point A(2, 1, 1) is perpendicular to the line segment joining the points B(3, 5, -1) and C(4,3, -1).

Find the value of p for which the line through the points A(3, 5, -1) and B(5, p, 0) 9 is perpendicular to the line through the points C(2, 1, 1) and D (3, 3, -1).

If O is the origin and P (2, 3,4) and Q (1, -2, 1) be any two points show that OPOQ.

Show that the line segment joining the points A(1, 2, 3) and B (4, 5, 7) is parallel to the segment joining the points C(-4, 3, -6) and D (2, 9, 2).

If the line segment joining the points A(7, p, 2) and B(q, -2, 5) be parallel to the line segment joining the points C(2, -3, 5) and D(-6, -15,11), find the values of p and q.

Show that the points A(2, 3, 4), B(-1, -2, 1) and C (5, 8, 7) are collinear.

Show that the points A(-2, 4.7), B(3, -6. -8) and C(1, -2, -2) are collinear.

Find the value of p for which the points A(-1, 3, 2), B(-4, 2, -2), and C(5, 5, p) are collinear.

Find the angle between the two lines whose direction cosines are:

and

Find the angle between the two lines whose direction ratios are:

a, b, c and (b – c), (c – a), (a – b).

Find the angle between the lines whose direction ratios are:

2, -3, 4 and 1, 2, 1.

1, 1, 2 and ,4

Find the angle between the vectors and

Find the angles made by the following vectors with the coordinate axes:

(i)

(ii)

(iii)

Find the coordinates of the foot of the perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, -1, 3) and C(2, -3, -1).