Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

Find the scalar components and magnitude of the vector joining the points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂).

A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.

Find the value of x for which is a unit vector.

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors and

If , and , find a unit vector parallel to the vector .

Show that the points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7) are collinear, and find the ratio in which B divides AC.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are externally in the ratio 1 : 2. Also, show that P is the midpoint of the line segment RQ.

The two adjacent sides of a parallelogram are and . Find the unit vector parallel to its diagonal. Also, find its area.

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are

.

Let and . Find a vector which is perpendicular to both a and , and

The scalar product of the vector a unit vector along the sum of vectors and is equal to one. Find the value of λ.

If are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to and

Prove that , if and only if are perpendicular, given .

If is the angle between two vectors and , then only when

Let and be two unit vectors and q is the angle between them. Then is a unit vector if

The value of is

If θ is the angle between any two vectors and , then when θ is equal to