Dot Product in Terms of Cosine

Let $\vec{a}=\left< a_1,a_2 \right>$, $\vec{b}=\left< b_1,b_2 \right>$, and $\vec{c}=\left< c_1,c_2 \right>$ be three vectors in two-dimensional space.

From the definition of the dot product, $$\vec{a}\cdot\vec{b}=a_1b_1+a_2b_2$$

We will now prove its equivalence to an equally useful geometric definition.

Note the following facts: $$\vec{a}\cdot\vec{a}=a_1^2+a_2^2=|\vec{a}|^2$$ $$\vec{a}\cdot(\vec{b}+\vec{c})=a_1(b_1+c_1)+a_2(b_2+c_2)=(a_1b_1+a_2b_2)+(a_1c_1+a_2c_2)=\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}$$

Notice that the dot product of a vector with itself is its magnitude squared, and that the dot product obeys the distributive property.Of course, the dot product is also commutative.

Consider a triangle $OAB$, where $\vec{OA}=\vec{a}$ and $\vec{OB}=\vec{b}$. Then $\vec{AB}=\vec{b}-\vec{a}$. Let $\theta$ be the measure of angle $AOB$.

Computing $AB^2$ using the Law of Cosines, $$AB^2=OA^2+OB^2-2OA\cdot OBcos\theta$$

Substituting in vectors and using the fact that $\vec{a}\cdot\vec{a}=|\vec{a}|^2$, $$(\vec{b}-\vec{a})\cdot(\vec{b}-\vec{a})=\vec{a}\cdot\vec{a}+\vec{b}\cdot\vec{b}-2|\vec{a}||\vec{b}|cos\theta$$

Using the distributive property, $$\vec{b}\cdot\vec{b}-2\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{a}=\vec{a}\cdot\vec{a}+\vec{b}\cdot\vec{b}-2|\vec{a}||\vec{b}|cos\theta$$

Cancelling and simplifying, $$\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|cos\theta$$

The dot product of two vectors is the product of their magnitudes and the cosine of the angle between them.

This is useful when one would like to compute the angle between two vectors. The cosine of the angle will equal the dot product of the vectors divided by the product of their magnitudes.

There are two ways to solve the dot product of two vectors, one of which involves using the cosine value of the angle between the two vectors. If $a$ and $b$ are two vectors, and $|a|$ and $|b|$ are their magnitudes then the formula is as follows:

$a * b = |a| * |b|cos (theta)$

You can use this formula to find out the dot product of these two vectors. Let's try a simple practice problem.

Hope you understood my topic explanation! Please let me know if I have made any typos or mistakes.