Consider the vectors Bold uuequals=left angle 4 comma 0 right angle4,0 and Bold vvequals=left angle negative 3 comma negative 3

Question

3 years ago

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Consider the vectors Bold uuequals=left angle 4 comma 0 right angle4,0 and Bold vvequals=left angle negative 3 comma negative 3

right angle−3,−3. Sketch the vectors, find the angle between the vectors, and compute the dot product using the definition Bold u times Bold v equals StartAbsoluteValue Bold u EndAbsoluteValue StartAbsoluteValue Bold v EndAbsoluteValue cosine thetau•v=u vcosθ.

Mathematics

2 answers:

Ber [7] · Answer 1 · 2022-01-15T08:12:25+03:00

Answer:

Check the graph below for the sketch.

$\theta=135^{\circ}$

Step-by-step explanation:

1) Let's organize the data from these Latex codes.

$\vec{\mathbf{u}}=\left \langle 4,0 \right \rangle\:and\: \vec{\mathbf{v}}=\left \langle -3,-3 \right \rangle$

2) Sketching them (Check below). Notice that we are going to use the coordinates of each vector.

3) To find the angle between the vectors, we need to remember the Theorem:

If there is an angle between two vectors a and b, then we can calculate its angle using this relation:

$a*b=\left \| a \right \|\left \| b \right \|*cos\theta$

3.1) Then we need the Dot product between u and v. The Dot product is going to give us a scalar value for a product between vectors.

We will also need the norm of each vector. The norm will tell us the length of each one.

$\vec{u}*\vec{v}=\vec{u}*\vec{v}=4(-3)+0(-3)=-12\\\left \| u \right \|=\sqrt{4^2+0^2 }= \left \| u \right \|=4, \\\left \| v \right \|=\sqrt{(-3)^2+(-3)^2}, \left \| v \right \|=18=3\sqrt{2} \left\\$

3.2)

Now, we can plug these pieces of information from 3.1 and 3.2 and find the angle:

$cos\theta=\frac{uv}{\left \| u \right \|\left \| v \right \|}=\frac{4*-3+(0)(-3)}{12\sqrt{2} }=\frac{-12}{ 12\sqrt{2} }\\\theta=cos^{-1}(\frac{-12}{ 12\sqrt{2} })\rightarrow \theta\approx 135^{\circ}$