Question

multiple choice question
for vector a, rotated counterclockwise through an angle of θ to the positive x - axis, the components of this vector can be expressed as
o ( a _ { x } = a sin \theta ) and ( a _ { y } = a cos \theta )
o ( a _ { x } = a cos \theta ) and ( a _ { y } = a \tan \theta )
o ( a _ { x } = a \tan \theta ) and ( a _ { y } = a sin \theta )
o ( a _ { x } = a cos \theta ) and ( a _ { y } = a sin \theta )

Explanation:

Step1: Recall vector - component formula

For a vector \(\vec{A}\) with magnitude \(A\) and angle \(\theta\) with the positive \(x -\)axis, we use the right - triangle trigonometry.
If we consider the vector \(\vec{A}\) as the hypotenuse of a right - triangle, the \(x -\)component \(A_{x}\) and \(y -\)component \(A_{y}\) are the adjacent and opposite sides respectively with respect to the angle \(\theta\).
The cosine of an angle \(\theta\) in a right - triangle is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), and the sine of an angle \(\theta\) is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\).

Step2: Calculate \(x\) and \(y\) components

If the magnitude of the vector is \(A\), then \(A_{x}=A\cos\theta\) (since \(A_{x}\) is the adjacent side to the angle \(\theta\) and \(A\) is the hypotenuse) and \(A_{y}=A\sin\theta\) (since \(A_{y}\) is the opposite side to the angle \(\theta\) and \(A\) is the hypotenuse).

Let's check each option:

• Option 1: \(A_{x}=A\sin\theta\) and \(A_{y}=A\cos\theta\) is incorrect. Using the right - triangle trigonometry definitions (\(x\) - component should be related to \(\cos\theta\) and \(y\) - component to \(\sin\theta\) for an angle with the \(x -\)axis).
• Option 2: \(A_{y}=A\tan\theta\) is incorrect. Since \(A_{y}=A\sin\theta\) and \(A_{x}=A\cos\theta\), \(\tan\theta=\frac{A_{y}}{A_{x}}\), not \(A_{y}=A\tan\theta\).
• Option 3: \(A_{x}=A\tan\theta\) is incorrect. \(A_{x}=A\cos\theta\) from the right - triangle trigonometry.
• Option 4: \(A_{x}=A\cos\theta\) and \(A_{y}=A\sin\theta\) is correct.

Answer:

\(A_{x}=A\cos\theta\) and \(A_{y}=A\sin\theta\) (the fourth option)