1.4 The Dot Product

The Dot Product

Introduction

The Dot Product

The Angle between Vectors

In this activity, we will use the following vectors:

Draw representatives of each of $\vu_1,\vu_2,\vu_3,\vu_4$ in standard position using the axes below.

Work through this exercise and explain your reasoning step by step.

Let $\vu,\vv,\vw$ be vectors in $\R^n$. Suppose that you know that $\vu\cdot \vw = 10$ and $\vv\cdot \vw = -3$. Compute $(\vu+\vv)\cdot \vw$.

Let $\vu,\vv$ be vectors in $\R^n$. Suppose that you know that $\vu\cdot \vv = 3$, $\vecmag{\vu} = 4$, and $\vecmag{\vv} = 7$. Compute $(\vu+\vv)\cdot (\vu+\vv)$.

Let $\vu,\vv$ be vectors in $\R^n$. For what value of $t$ is $(t\vu + \vv) \cdot \vv =0$?

The length of the vector $\vu=\langle 1, 2, -3 \rangle$ using the dot product.

The angle between the vectors $\vu =\langle 1, 2 \rangle$ and $\vv = \langle 4, -1 \rangle$ to the nearest tenth of a degree.

The angle between the vectors $\vy =\langle 1, 2, -3 \rangle$ and $\vz = \langle -2, 1, 1 \rangle$ to the nearest tenth of a degree.

If the angle between the vectors $\vu$ and $\vv$ is a right angle, what does the expression $\vu \cdot \vv = \vecmag{\vu} \vecmag{\vv} \cos(\theta)$ say about their dot product?

If the angle between the vectors $\vu$ and $\vv$ is acute——that is, less than $\pi/2$——what does the expression $\vu\cdot\vv=\vecmag{\vu}\vecmag{\vv}\cos(\theta)$ say about their dot product?

If the angle between the vectors $\vu$ and $\vv$ is obtuse——that is, greater than $\pi/2$——what does the expression $\vu\cdot\vv=\vecmag{\vu}\vecmag{\vv}\cos(\theta)$ say about their dot product?