If the vectors and have magnitudes of 10 and 11, respectively, and the scalar product of these two vectors is −100, what is the magnitude of the sum of these two vectors?
A) 6.6
B) 4.6
C) 8.3
D) 9.8
E) 7.6

Question

If the vectors  and  have magnitudes of 10 and 11, respectively, and the scalar product of these two vectors is −100, what is the magnitude of the sum of these two vectors?
A) 6.6 
B) 4.6 
C) 8.3 
D) 9.8 
E) 7.6

Quizplus · Accepted Answer

Let the two vectors be $\vec{a}$ and $\vec{b}$. We are given that $|\vec{a}|=10$ and $|\vec{b}|=11$. We are also given that $\vec{a} \cdot \vec{b}=-100$. Using the fact that $\vec{a} \cdot \vec{b} = |\vec{a}| \cdot |\vec{b}| \cdot \cos{	heta}$, we can find $\cos{	heta} = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \cdot |\vec{b}|} = -\frac{100}{110}=-\frac{10}{11}$.

Now, let $\vec{c}=\vec{a}+\vec{b}$. We want to find $|\vec{c}|$. Using the Law of Cosines, we have $|\vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|\cos{	heta}$. Substituting the given values, we get $|\vec{c}|^2 = 10^2 + 11^2 + 2 \cdot 10 \cdot 11 \cdot \left(-\frac{10}{11}ight)=4$. Therefore, $|\vec{c}|=\boxed{	extbf{(B) }4.6}$.

If the Vectors and Have Magnitudes of 10