Question

Find the angle between the vectors. U=< -4,-3> V = < -1,5>

Answer 1

Step-by-step explanation:

The formula for the angle between vectors

$\alpha = \cos {}^{ - 1} ( \frac{uv}{ |u| |v| } )$

To multiply vectors, multiply the first component and multiply the second component and add them.

(-4*-1) + (-3*5)= 4-15=-11.

To find magnitude of vectors, use the Pythagorean theorem

$u = \sqrt{ { - 4}^{2} + { - 3}^{2} } = 5$

$v = \sqrt{ - 1 {}^{2} + 5 {}^{2} } = \sqrt{26}$

so

$|u| |v| = 5 \sqrt{26}$

Know we have,

$\alpha = \cos {}^{ - 1} ( \frac{7}{5 \sqrt{26} } )$

$\alpha = 105.94$

in degrees,

$\alpha = 1.849$

in radians

Answer 2

Answer:

115.6° (1 d.p.)

Step-by-step explanation:

To find the angle between two vectors:

• Create a triangle with the vectors as two sides and the included angle θ between them.
• Find the magnitude of each vector (the length of each side of the triangle).
• Use the cosine rule to find the angle θ.

**Please see attached for the triangle diagram**

Given vectors:

$\textbf{u}=-4\textbf{i}-3\textbf{j}$

$\textbf{v}=-\textbf{i}+5\textbf{j}$

Use Pythagoras Theorem to find the magnitude of each vector:

$\implies |\textbf{u}|=\sqrt{(-4)^2+(-3)^2}=5$

$\implies |\textbf{v}|=\sqrt{(-1)^2+5^2}=\sqrt{26}$

$\overrightarrow{\text{UV}}=\textbf{v}-\textbf{u}=(-\textbf{i}+5\textbf{j})-(-4\textbf{i}-3\textbf{j})=3\textbf{i}+8\textbf{j}$

$|\overrightarrow{\text{UV}}|=\sqrt{3^2+8^2}=\sqrt{73}$

Cosine Rule (for finding angles)

$\sf \cos(C)=\dfrac{a^2+b^2-c^2}{2ab}$

where:

• C = angle
• a and b = sides adjacent the angle
• c = side opposite the angle

Find angle θ using the cosine rule:

$\implies \cos(\theta)=\dfrac{|\textbf{u}|^2+|\textbf{v}|^2-|\overrightarrow{\text{UV}}|^2}{2|\textbf{u}||\textbf{v}|}$

$\implies \cos(\theta)=\dfrac{5^2+\left(\sqrt{26}\right)^2-\left(\sqrt{73}\right)^2}{2(5)\left(\sqrt{26}\right)}$

$\implies \cos(\theta)=\dfrac{-22}{10\sqrt{26}}$

$\implies \theta=\cos^{-1}\left(\dfrac{-22}{10\sqrt{26}}\right)$

$\implies \theta=115.5599652...^{\circ}$

Therefore, the angle between the vectors is 115.6° (1 d.p.).