Question

# Vectors V_1 and V_2 are different vectors with lengths V1 and V2 respectively.

Vectors

Vectors $$V_1$$ and $$V_2$$ are different vectors with lengths $$V_1$$ and $$V_2$$ respectively. Find the following:
a) $$V_1\cdot V_1$$ Express you answer in terms of $$V_1$$
b) $$V_1\cdot V_2$$, when they are perpendicular
c) $$V_1\cdot V_2$$, when they are parallel

2021-06-05

Dot product of two vectors is equal to the magnitude of first vector multiplied by magnitude of second vector multiplied by the cosine angle between the two vectors.
$$\vec{A}\cdot\vec{B}=|\vec{A}||\vec{B}|\cos\theta$$
a) $$\vec{V_1}\cdot\vec{V_1}=|\vec{V_1}|\cdot|\vec{V_1}|\cos0^\circ$$
$$=(V_1)(V_1)$$
$$=V_1^2$$
b) $$\vec{V_1}\cdot\vec{V_2}=|\vec{V_1}|\cdot|\vec{V_2}|\cos\theta$$
If the two vectors are perpendicular to each other, then $$\theta=90^\circ$$
Therefore,
$$\vec{V_1}\cdot\vec{V_2}=|\vec{V_1}|\cdot|\vec{V_2}|\cos90^\circ$$
$$=(V_1)(V_2)(0)$$
$$=0$$
d) $$\vec{V_1}\cdot\vec{V_2}=|\vec{V_1}|\cdot|\vec{V_2}|\cos\theta$$
If the two vectors are parallel, the $$\theta=0^\circ$$
Therefore
$$\vec{V_1}\cdot|\vec{V_1}|\cdot|\vec{V_2}|\cos0^\circ$$
$$=(V_1)(V_2)(1)$$
$$=V_1V_2$$