How to Find Vector Components?

How to Find Vector Components? Is a multiple-choice math challenge from Effortless Math. Read the problem, choose from the answer options below, then open the solution to see the full step-by-step working and check your reasoning against it.

• Trigonometric ratios show the relationship between the magnitude of the vector and the components of the vector.

• Using “Pythagoras theorem” in right triangles with lengths (v_x) and (v_y):

Vector Components, Example 1:

(color{blue}{v_x= v cos θ})

(color{blue}{v_y= v sin θ})

(v_x= vcos 60° ) → (v_x= 20×frac{1}{2}= frac{20}{2}=10)

(v_y= vsin 60° ) → (v_y= 20×frac{sqrt{3}}{2}= frac{20sqrt{3}}{2} =10sqrt{3})

So, the vector (v⃗) is ((10, 10sqrt{3})).

Vector Components, Example 2:

Find the (x) and (y) components of a vector having a magnitude of (10) and make an angle of (45) degrees with the positive (x)-axis.

To find the components of a vector use these formulas:

(color{blue}{v_x= v cos θ})

(color{blue}{v_y= v sin θ})

(v_x= vcos 45° ) → (v_x= 10×frac{sqrt{2}}{2}= frac{10sqrt{2}}{2} =5sqrt{2})

(v_y= vsin 45° ) → (v_y= 10×frac{sqrt{2}}{2}= frac{10sqrt{2}}{2} =5sqrt{2})

So, the (x)-component and the (y)-components of the vector are both equal to (5sqrt{2}).

Exercises for Vector Components

1. Find the value of ( θ ), if (v_x=15) and (v_y=8.66).
2. Find out the magnitude of a vector (OA=(-3,4)).
3. Find the components of the vector, if the magnitude of a vector (v⃗) is (6) units and the direction of the vector is (30°) with the horizontal.
4. Find the direction of ((-4,3)).

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1. (color{blue}{θ=30^circ})
2. (color{blue}{|OA|=5})
3. (color{blue}{v⃗=3, 3sqrt{3}})
4. (color{blue}{θ=143.13^circ})

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