I.6. Decomposition of a vector.

• I.6.1. Decomposition of a vector along two mutually perpendicular directions.
• I.6.2. Decomposition of a vector along two given directions.

I.6.1. Decomposition of a vector along two mutually perpendicular directions.

The decomposition of a vector along two mutually perpendicular directions is done as follows:

• From the top of the given vector we go perpendicular on the two directions Ox and Oy.
• We write the vectorial equation:

• We write the scalar equation:

🔦 Remark

The decomposition of a vector in two mutually perpendicular directions is the opposite of the composition of two nonlinear and with the same application point vectors, with the parallelogram rule, specifying that the component vectors are in the two main directions: F_x horizontally and F_y vertically. Thus we obtain a rectangle whose sides are the segments of the component forces.

🔓 Solved problem

1. Laurence strikes a nail with a hammer with a force of 500 N in a wall, holding the nail inclined towards the wall with an angle α = 38°. What is the value of the forces that compose Laurence's force?

Solution:

F = 500 N, direction that makes an angle of 38° with the vertical.

We can find out the two forces by the graphical method.

Standard: 1 cm : 100 N

500 N : 100 N = 5 cm represents the segment of the force F and we draw it.

From the top of the vector F they go perpendicular to the two directions Ox and Oy.

We measure the segments of the component vectors with the ruler and we multiply with the standard to find out their values.

Fx = 3 ∙ 100 = 300 N
Fy = 4 ∙ 100 = 400 N

We write the vectorial equation:

We check with Pythagoras' theorem

We write the scalar equation:

500² = 300² + 400²

250.000 = 90.000 + 16.0000

I.6.2. Decomposition of a vector along two given directions.

Decomposition of a vector along two given directions Δ1 și Δ2 it is done as follows:

• From the top of the vector go parallel to the two given directions.

• We write the vectorial equation:

• We write the scalar equation:

🔦 Remark

The decomposition of a vector along two given directions is the opposite of the composition of two nonlinear and with the same application point vectors, with the parallelogram rule.