Questions: The magnitudes of vectors u and v and the angle θ between the vectors are given. Find the sum of u+v. u=13, v=13, θ=110° The magnitude of u+v is . (Round to the nearest tenth as needed.) The resultant vector u+v makes an angle of ° with u (Round to the nearest degree as needed.)

Solution Steps

To find the magnitude of the sum of two vectors \(\mathbf{u}\) and \(\mathbf{v}\), we use the formula for the magnitude of the resultant vector:

\[ |\mathbf{u} + \mathbf{v}| = \sqrt{|\mathbf{u}|^2 + |\mathbf{v}|^2 + 2 |\mathbf{u}| |\mathbf{v}| \cos \theta} \]

Given:

• \(|\mathbf{u}| = 13\)
• \(|\mathbf{v}| = 13\)
• \(\theta = 110^\circ\)

Substitute these values into the formula:

\[ |\mathbf{u} + \mathbf{v}| = \sqrt{13^2 + 13^2 + 2 \times 13 \times 13 \times \cos 110^\circ} \]

Calculate \(\cos 110^\circ\):

\[ \cos 110^\circ \approx -0.3420 \]

Substitute \(\cos 110^\circ\) into the equation:

\[ |\mathbf{u} + \mathbf{v}| = \sqrt{169 + 169 + 2 \times 13 \times 13 \times (-0.3420)} \]

\[ = \sqrt{338 - 115.668} \]

\[ = \sqrt{222.332} \]

\[ \approx 14.9 \]

To find the angle \(\phi\) that the resultant vector \(\mathbf{u} + \mathbf{v}\) makes with \(\mathbf{u}\), we use the formula:

\[ \tan \phi = \frac{|\mathbf{v}| \sin \theta}{|\mathbf{u}| + |\mathbf{v}| \cos \theta} \]

Substitute the known values:

\[ \tan \phi = \frac{13 \times \sin 110^\circ}{13 + 13 \times (-0.3420)} \]

Calculate \(\sin 110^\circ\):

\[ \sin 110^\circ \approx 0.9397 \]

Substitute \(\sin 110^\circ\) into the equation:

\[ \tan \phi = \frac{13 \times 0.9397}{13 - 4.446} \]

\[ = \frac{12.2161}{8.554} \]

\[ \approx 1.4287 \]

Calculate \(\phi\):

\[ \phi \approx \tan^{-1}(1.4287) \approx 55^\circ \]

Final Answer