Questions: Given the vectors u=6i+5j and v=7i-7j, find the angle (in degrees) between them. Angle between u and v= - Round answers to 2 decimal places as needed.

Given the vectors u=6i+5j and v=7i-7j, find the angle (in degrees) between them.
Angle between u and v= -
Round answers to 2 decimal places as needed.

Transcript text: Given the vectors $\vec{u}=6 i+5 j$ and $\vec{v}=7 i-7 j$, find the angle (in degrees) between them. Angle between $\vec{u}$ and $\vec{v}=$ $\square$ - Round answers to 2 decimal places as needed.

Solution

Solution Steps

Step 1: Calculate the dot product of u and v

The dot product of ^u^ and ^v^ is calculated as $\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 = 6 \times 7 + 5 \times -7 = 7$.

Step 2: Calculate the magnitudes of u and v

The magnitude of ^u^ is $\|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2} = \sqrt{6^2 + 5^2} = 7.810$, and the magnitude of ^v^ is $\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2} = \sqrt{7^2 - 7^2} = 9.899$.

Step 3: Use the dot product and magnitudes to find the cosine of the angle theta

The cosine of the angle $\theta$ is calculated as $\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|} = \frac{7}{7.810 \times 9.899} = 0.0905$.

Step 4: Calculate the angle theta by taking the inverse cosine (arccos) of the result from step 3

The angle $\theta$ in radians is $\arccos(0.0905) = 1.480$ radians.

Step 5: Convert theta from radians to degrees

The angle $\theta$ in degrees is $\theta \times \frac{180}{\pi} = 84.806$ degrees.