The points A, B and C have coordinates (1, 0), (3, 7) and (3, 5) respectively. Find $\angle BAC$ . Answer ___

It won't just hand you the answer — it asks where you're stuck.How the checker works

Worked solution

1. Find the vectors representing the two arms of $\angle BAC$.

Since $\angle BAC$ is the angle between $AB$ and $AC$ ,

\overrightarrow{AB}=\begin{pmatrix}3-1\\7-0\end{pmatrix}=\begin{pmatrix}2\\7\end{pmatrix},\qquad \overrightarrow{AC}=\begin{pmatrix}3-1\\5-0\end{pmatrix}=\begin{pmatrix}2\\5\end{pmatrix}.

2. Find the magnitudes of the two vectors.

|\overrightarrow{AB}|=\sqrt{2^2+7^2}=\sqrt{53},\qquad |\overrightarrow{AC}|=\sqrt{2^2+5^2}=\sqrt{29}.

3. Apply the scalar product formula to find the cosine of the angle.

Let $\theta=\angle BAC$ .

$\overrightarrow{AB}\cdot\overrightarrow{AC}=2(2)+7(5)=39.$

Using $\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta$ ,

$\cos\theta=\frac{39}{\sqrt{53}\sqrt{29}}.$

4. Evaluate the angle using inverse cosine.

$\theta=\cos^{-1}\left(\frac{39}{\sqrt{53}\sqrt{29}}\right)$

$\theta\approx 29.7^\circ.$

Answer

$29.7^\circ$

AI analysis — may contain mistakes.

Worked this one out on paper? Snap your steps and Gigle's AI math checker will circle the step where it went wrong.