Home > PC3 > Chapter 11 > Lesson 11.2.1 > Problem11-94

11-94.

Given vector $\vec { v }= ⟨4, 3⟩$ and vector $\vec { u }= ⟨a, 5⟩$, what must the value of a be in order for the angle between the two vectors to equal $60°$?

Dot Product

Given two vectors $\vec{\text{u}}=⟨\text{u}1,\text{u}2⟩$ and $\vec { \text{v} }= ⟨\text{v}1, \text{v}2⟩$ , the dot product (also called inner product or scalar product) is:

$\vec{\text{u}}·\vec{\text{v}}=\text{u}_1\text{v}_1+\text{u}_2\text{v}_2$

Example: If $\vec { \text{u} }= ⟨–2, 7⟩$ and $\vec { \text{v} }= ⟨3, 4⟩$ , then $\vec { \text{u} } · \vec{\text{v}}= –2(3) + 7(4) = 22$.

Another formula for the dot product is:

$\vec { \text{u} } \cdot \vec { \text{v} } = \| \text{u} \| \| \text{v} \| \cos ( \theta )$, where θ is the angle between $\vec { \text{u} }$ and $\vec { \text{v} }$

$\cos(60º)=\frac{4a+3(5)}{\left( \sqrt{3^2+4^2} \right)\left( \sqrt{a^2+5^2} \right) }$