Given two unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ on the unit circle, what is the angle $\theta$ between them if $\overrightarrow{a}$ = $\left(1,0\right)$ and $\overrightarrow{b}$ = $\left(0,1\right)$? Express your answer using one significant figure.

• A. $90°$
• B. $60°$
• C. $45°$
• D. $180°$

Answer: (A)

Verified step by step guidance

1. Identify the given vectors \( \vec{a} = (1, 0) \) and \( \vec{b} = (0, 1) \), both of which lie on the unit circle, meaning their magnitudes are 1.
2. Recall the formula for the angle \( \theta \) between two vectors using the dot product: \[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \]
3. Calculate the dot product of \( \vec{a} \) and \( \vec{b} \): \[ \vec{a} \cdot \vec{b} = (1)(0) + (0)(1) = 0 \]
4. Since both vectors are unit vectors, their magnitudes are 1, so substitute into the dot product formula: \[ 0 = 1 \times 1 \times \cos(\theta) \implies \cos(\theta) = 0 \]
5. Find the angle \( \theta \) by taking the inverse cosine (arccos) of 0: \[ \theta = \arccos(0) \]. This angle corresponds to one of the standard angles on the unit circle.