Operations with Vectors.

 

Two vectors \( v = \langle 1, 3 \rangle \) and \( w = \langle 2, 2 \rangle \) are given.
(a) What angle does v make with the x-axis?
(b) Find a vector u such that u + v = -w

The angle between a vector and the x-axis is the gradient of the vector

If \( \theta \) is the angle then \[ \begin{align} \tan \theta &= \frac{v_2}{v_1} \\ &= \frac{3}{1} \end{align} \ \] and the angle is \[ \begin{align} \theta &= \tan^{-1}(\frac{3}{1}) \\ &= 1.25 \quad \text{radians} \end{align} \ \]

Use components

\[ \begin{align} u + v &= -w \\ \end{align} \ \] in components this is \[ \begin{align} \langle u_1, u_2 \rangle + \langle 1, 3 \rangle &= -\langle 2, 2 \rangle \\ \end{align} \ \]

Solve for the components of u

\[ \begin{align} u_1 + 1 &= -2 \\ u_1 &= -3 \end{align} \ \] and \[ \begin{align} u_2 + 3 &= -2 \\ u_2 &= -5 \end{align} \ \] then \[ \begin{align} u &= \langle -3, -5 \rangle\\ \end{align} \ \]

Important This is not a 'textbook on a screen' - this is an interactive learning system. Which means you have to 'act' to make it go.

Look through the options below and select the one you think is the best response. You will gain most benefit if you work the problem with pen and paper as you go, i.e. decide what you think is the correct option and write down what you think the result will be before you click. If the next step doesn't display, click another option.

 

What is the relationship between a vector and its angle with the x-axis?

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Now You Try:

Two vectors \( v = \langle 1, -3 \rangle \) and \( w = \langle -3, -2 \rangle \) are given.
(a) What angle does v make with the x-axis?
(b) Find a vector u such that u + v + w = 0

 

The angle is the gradient of the vector
The sine of the angle is the ratio of the components
The tan of the angle is the ratio of the components