How do you find the x and y components of a vector?

What "resolving" a vector means

Earlier in this chapter, two vectors were added to get one. Resolution runs the other way. You take one vector and split it into two that add up to it, along directions you choose. Pick the x and y axes, and every problem in the plane turns into two straight-line problems. That is why resolution matters, and you will lean on it for projectiles and circular motion.

Unit vectors come first

NCERT introduces î, ĵ and k̂: unit vectors along the x, y and z axes. A unit vector has magnitude 1 and only points a direction. It has no unit and no dimension.

Write this correctly in the exam: |î| = 1. Not î = 1. The left side is a magnitude, a plain number. î itself is a direction arrow and can't equal a number.

With them, any vector in the x-y plane can be written as

A = A_x î + A_y ĵ

A sentence from the book worth copying into your notes: "A_x is itself not a vector, but A_x î is a vector." A_x is just a number with a sign and a unit. Multiplying it by î is what gives it a direction.

The formulas, and why θ must start at the x-axis

Drop a perpendicular from the tip of A to each axis. You get a right triangle with the vector as hypotenuse. The side along x is adjacent to θ, so it takes the cosine. The side along y is opposite, so it takes the sine:

A_x = A cos θ A_y = A sin θ

This only works if θ is measured from the positive x-axis. Many problems don't hand it to you that way. In the umbrella example from the previous lesson, the angle was measured from the vertical. With θ taken from the vertical, the vertical component becomes R cos θ and the horizontal one R sin θ. Sketch the triangle each time instead of trusting the formula blindly.

Components can be negative, or zero

Take the same 10 m displacement but now at 120° from the positive x-axis. cos 120° = −0.5, so A_x = −5 m. The minus sign just says the component points along −x. Meanwhile A_y = 10 sin 120° ≈ 8.7 m. A vector lying along the y-axis has A_x = 0 exactly.

Going backwards: magnitude and direction from components

Given A_x and A_y, you can rebuild the vector:

A = √(A_x² + A_y²) tan θ = A_y / A_x

The square-root formula is Pythagoras, and it works only because the x and y axes are perpendicular.

The tan formula needs care. Compare (3, 4) and (−3, −4). Both give tan θ = 4/3, yet the first vector points into the first quadrant and the second into the third, exactly opposite. Your calculator returns 53° for both. Check the signs of A_x and A_y and decide the quadrant yourself.

A note on choosing directions

NCERT also shows that a vector can be split along any two directions a and b, so long as they are non-zero and not along the same line. They don't have to be at right angles. Perpendicular axes are chosen because they make the arithmetic clean: Pythagoras works and each component is a simple cos or sin.

In three dimensions, add a third term: A = A_x î + A_y ĵ + A_z k̂. The position vector of a point (x, y, z) is r = x î + y ĵ + z k̂.

Check yourself

A trolley is pulled with a force of 40 N at 60° above the horizontal floor. What are the horizontal and vertical components? (Answer at the bottom.)

Questions students ask

Answer: horizontal 40 cos 60° = 20 N, vertical 40 sin 60° ≈ 34.6 N.