If a = i + j + k, b = 2i − j + 3k and c = i − 2j + k, find a unit vector parallel to the vector 2a – b + 3c.

Find |a| and |b|, if (a + b) dot product (a - b) = 8 and |a| = 8 |b|.

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors a = 2i + 3j −k and b = i − 2j + k.

Given that a b 0 and a x b = 0. What can you conclude about the vectors a and b ?

Find the sum of the vectors a = i − 2j + k, b = −2i + 4j + 5k and c = i − 6j – 7k.

Find lambda and mu if (2i + 6j + 27k) x (i + lambda j + mu k) = vector 0.

Show that each of the given three vectors is a unit vector:

1/7 (2i + 3j + 6k), 1/7 (3i - 6j + 2k), 1/7(6i + 2j - 3k)

Also show that they are mutually perpendicular to each other.

Answer the following as true or false.

(i) a vector and −a vector are collinear.

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (– 5, 7).

Find the value of x for which x(i + j + k) is a unit vector.

Find the projection of the vector i + 3j + 7k on the vector 7i - j + 8k.

Find the values of x and y so that the vectors 2i + 3j and xi + yj are equal.

Show that (a - b) x (a + b) = 2(a x b).

If a = b + c, then is it true that |a| = |b| + |c|? Justify your answer.