3D representation of vector

By Admin Koushi
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March 26, 2025

3D representation of vector

We can use 3D representation of a vector to calculate position vector of a point. Angle of vector with given axis can be calculated by 3D representation of vector also.

Representation of a vector by coordinates:

Let us consider, OX, OY and OZ are three perpendicular axes, where O is the origin. Let P is a point with coordinates (x, y, z) and = is the position vector of P. , and are the unit vectors along + ve X, Y and Z axes. Let PB is the perpendicular drawn on x-y plane. BA and BC are the perpendicular drawn on X and Y axis respectively. PD is the perpendicular drawn on Z axis.

Therefore = x, = = y and = = z and their magnitudes are OA = x, AB = OC = y and BP = OD = z.

Or, = + +

= x + y + z.

Magnitude: From OAB, OB² = OA² + AB² and from OBP, OP² = OB² + BP²

From OAB, = + and from OBP, = +

Or, OP² = OA² + AB² + BP²

So, R² = x² + y² + z²

R =

Direction: Let us consider, , and are the angles of R with +ve X, Y and Z axes respectively. Then, cos = , cos = , cos = where cos, cos and cos are the direction cosines of .

Problem on representation of a vector by coordinates:

1. A particle moves with constant speed v on the circumference of a circle. Find the velocity of the particle at points A, B and C.

At point A, the direction of velocity of the particle is along positive Y axis. So, = v .

At point B, the velocity of the particle creates angle with positive X axis. So, = vsin(-) + vcos.

At point C, the velocity of the particle is along negative X-axis. So, = v (-).

2. The coordinates of two points P and Q are respectively (, , ) and (, , ) then find .

The position vector of point P is = î + ĵ + k̂ and the position vector of point Q is = î + ĵ + k̂.

Using triangle law of vector addition = –

Or, = (î + ĵ + k̂) – (î + ĵ + k̂)

= ( – )î + ( – )ĵ + ( – )k̂.

= =

3. If (= 2î + nĵ – k̂) creates angle with positive y axis then what is the value of n?

Direction cosine cos = or, cos =

Or, =

n = .

4. Two-point charges q₁ and q₂ are placed at (0, 0, 0) and (1, 2, 2) m respectively. They repel each other with force of 9 N. The force on q₂ due to q₁ is = x + y + z. Find the value of x + y + z.