Find the angle $$ \theta $$ between the vectors. $$ \begin{array}{l}\mathbf{u}=6 \mathbf{i}+7 \mathbf{j}+\mathbf{k} \ \mathbf{v}=7 \mathbf{i}-6 \mathbf{j}\end{array} $$ $$ \theta=\square $$ radians

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UpStudy AI · Accepted Answer

To find the angle $$ \theta $$ between the vectors $$\mathbf{u}$$ and $$\mathbf{v}$$, we can use the dot product formula:

$$
\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|}
$$

**Step 1: Compute the Dot Product $$\mathbf{u} \cdot \mathbf{v}$$**

$$
\mathbf{u} = 6\mathbf{i} + 7\mathbf{j} + \mathbf{k}
$$
$$
\mathbf{v} = 7\mathbf{i} - 6\mathbf{j}
$$
$$
\mathbf{u} \cdot \mathbf{v} = (6)(7) + (7)(-6) + (1)(0) = 42 - 42 + 0 = 0
$$

**Step 2: Compute the Magnitudes $$|\mathbf{u}|$$ and $$|\mathbf{v}|$$**

$$
|\mathbf{u}| = \sqrt{6^2 + 7^2 + 1^2} = \sqrt{36 + 49 + 1} = \sqrt{86}
$$
$$
|\mathbf{v}| = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}
$$

**Step 3: Compute $$\cos \theta$$**

$$
\cos \theta = \frac{0}{\sqrt{86} \cdot \sqrt{85}} = 0
$$

**Step 4: Determine $$\theta$$**

Since $$\cos \theta = 0$$, the angle $$\theta$$ is:

$$
\theta = \frac{\pi}{2} \text{ radians}
$$

**Final Answer:**
$$
\theta = \frac{\pi}{2} \text{ radians}
$$
The angle $$ \theta $$ between the vectors is $$ \frac{\pi}{2} $$ radians.

Find the angle between the vectors.

radians

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