Find the horizontal and vertical components of the vector with the given length and direction, and write the vector in terms of the vectors i and j.

Question

Find the horizontal and vertical components of the vector with the given length and direction, and write the vector in terms of the vectors i and j.

|𝘃| = 20, 𝜃 = 30°

Answer 1

x = |v| cosθ

y = |v| sinθ

Answer 2

To find the horizontal and vertical components of a vector, you can use trigonometry. The horizontal component is found by multiplying the length of the vector by the cosine of the angle, and the vertical component is found by multiplying the length of the vector by the sine of the angle.

Let's use this information to find the horizontal and vertical components of the given vector with a length of 20 and an angle of 30°.

Horizontal component (𝑥):
𝑥 = |𝘃| × cos(𝜃)

Substituting the values:
𝑥 = 20 × cos(30°)

Using the values of cosine for 30 degrees (which is √3/2):
𝑥 = 20 × (√3/2)
= 20 × (√3/2)
= 20 × (√3/2)
= 20 × 0.866 ≈ 17.32

Therefore, the horizontal component of the vector is approximately 17.32.

Vertical component (𝑦):
𝑦 = |𝘃| × sin(𝜃)

Substituting the values:
𝑦 = 20 × sin(30°)

Using the values of sine for 30 degrees (which is 1/2):
𝑦 = 20 × (1/2)
= 20 × 0.5
= 10

Therefore, the vertical component of the vector is 10.

Now, let's write the vector in terms of the vectors i and j.

The horizontal component (𝑥) represents the force in the x-axis direction, or the component of the vector in the i direction. So, the horizontal component is 17.32 and can be written as (17.32)𝑖.

The vertical component (𝑦) represents the force in the y-axis direction, or the component of the vector in the j direction. So, the vertical component is 10 and can be written as (10)𝑗.

Therefore, the vector can be written as (17.32)𝑖 + (10)𝑗.