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Snezhnost [94]
3 years ago
11

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]3 years ago
6 0

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}  

saw5 [17]3 years ago
4 0

Answer:

Angle between the two vectors is 135°

u•v = -12

Step-by-step explanation:

Given two vectors u = (4,0) and v = (-3,-3).

To find the angle between the two vectors we will use the formula for calculating the angle between two vectors as shown;

u•v = |u||v|cos theta

cos theta = u•v/|u||v|

theta = arccos  (u•v/|u||v|)

u•v = (4,0)•(-3,-3)

u•v = 4(-3)+0(-3)

u•v = -12

For |u| and |v|

|u| = √4²+0²

|u| = √16 = 4

|v| = √(-3)²+(-3)²

|v| = √9+9

|v| = √18

|v| = 3√2

|u||v| = 4×3√2 = 12√2

theta = arccos(-12/12√2)

theta = arccos(- 1/√2)

theta = -45°

Since cos is negative in the second quadrant, theta = 180-45°

theta = 135°

To get u•v using the formula u•v = |u||v|cos theta

Given |u||v| = 12√2 and theta = 135°

u•v = 12√2cos 135°

u•v = 12√2× -1/√2

u•v = -12√2/√2

u•v = -12

For the diagram of the vectors, find it in the attachment below.

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<img src="https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B32%7D" id="TexFormula1" title="\sqrt[3]{32}" alt="\sqrt[3]{32}" align="absmid
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