Question

the magnitude of a vector v is 15 and its direction angle θ is 90°. write the component form for v. write each component in exact simplified form. v = ⟨ , ⟩

Explanation:

Step1: Recall the formula for component form

The component form of a vector \(v=\langle a,b
angle\) is given by \(a = \vert v\vert\cos\theta\) and \(b=\vert v\vert\sin\theta\), where \(\vert v\vert\) is the magnitude of the vector and \(\theta\) is the direction angle.

Step2: Calculate the \(x\) - component

Given \(\vert v\vert = 15\) and \(\theta = 90^{\circ}\).
For the \(x\) - component \(a\), we use \(a=\vert v\vert\cos\theta\).
Since \(\cos(90^{\circ})=0\), then \(a = 15\times\cos(90^{\circ})=15\times0 = 0\).

Step3: Calculate the \(y\) - component

For the \(y\) - component \(b\), we use \(b=\vert v\vert\sin\theta\).
Since \(\sin(90^{\circ}) = 1\), then \(b=15\times\sin(90^{\circ})=15\times1=15\).

Answer:

\(\langle0,15
angle\)