Questions: Find the magnitudes of the horizontal and vertical components for the vector v, if α is the direction angle of v from the horizontal α=39°, v=54 The magnitude of the horizontal component of v is (Round to the nearest integer as needed.)

Find the magnitudes of the horizontal and vertical components for the vector v, if α is the direction angle of v from the horizontal
α=39°, v=54

The magnitude of the horizontal component of v is
(Round to the nearest integer as needed.)

Transcript text: Find the magnitudes of the horizontal and vertical components for the vector $\mathbf{v}$, if $\boldsymbol{\alpha}$ is the direction angle of v from the horizontal \[ \boldsymbol{\alpha}=39^{\circ},|\mathbf{v}|=54 \] The magnitude of the horizontal component of $\mathbf{v}$ is $\square$ (Round to the nearest integer as needed.)

Solution

Solution Steps

Step 1: Convert the angle from degrees to radians

To convert the angle from degrees to radians, we use the formula $\alpha_{radians} = \frac{\pi \cdot \alpha_{degrees}}{180}$. For an angle of 39 degrees, this gives us $\alpha_{radians} = 1$.

Step 2: Calculate the horizontal component of the vector

The horizontal component is calculated using the formula $|\mathbf{v}| \cdot \cos(\alpha)$. Substituting the given values, we get $|\mathbf{v}| \cdot \cos(1) = 42$.

Step 3: Calculate the vertical component of the vector

The vertical component is calculated using the formula $|\mathbf{v}| \cdot \sin(\alpha)$. Substituting the given values, we get $|\mathbf{v}| \cdot \sin(1) = 34$.