Question

Find the angle between u =the square root of 5i-8j and v =the square root of 5i+j.

Answer 1

Answer:

The angle between vector $\vec{u} = 5\, \vec{i} - 8\, \vec{j}$ and $\vec{v} = 5\, \vec{i} + \, \vec{j}$ is approximately $1.21$ radians, which is equivalent to approximately $69.3^\circ$.

Step-by-step explanation:

The angle between two vectors can be found from the ratio between:

• their dot products, and
• the product of their lengths.

To be precise, if $\theta$ denotes the angle between $\vec{u}$ and $\vec{v}$ (assume that $0^\circ \le \theta < 180^\circ$ or equivalently $0 \le \theta < \pi$,) then:

$\displaystyle \cos(\theta) = \frac{\vec{u} \cdot \vec{v}}{\| u \| \cdot \| v \|}$.

Dot product of the two vectors

The first component of $\vec{u}$ is $5$ and the first component of $\vec{v}$ is also

The second component of $\vec{u}$ is $(-8)$ while the second component of $\vec{v}$ is $1$. The product of these two second components is $(-8) \times 1= (-8)$.

The dot product of $\vec{u}$ and $\vec{v}$ will thus be:

$\begin{aligned} \vec{u} \cdot \vec{v} = 5 \times 5 + (-8) \times1 = 17 \end{aligned}$.

Lengths of the two vectors

Apply the Pythagorean Theorem to both $\vec{u}$ and $\vec{v}$:

• $\| u \| = \sqrt{5^2 + (-8)^2} = \sqrt{89}$.
• $\| v \| = \sqrt{5^2 + 1^2} = \sqrt{26}$.

Angle between the two vectors

Let $\theta$ represent the angle between $\vec{u}$ and $\vec{v}$. Apply the formula$\displaystyle \cos(\theta) = \frac{\vec{u} \cdot \vec{v}}{\| u \| \cdot \| v \|}$ to find the cosine of this angle:

$\begin{aligned} \cos(\theta)&= \frac{\vec{u} \cdot \vec{v}}{\| u \| \cdot \| v \|} = \frac{17}{\sqrt{89}\cdot \sqrt{26}}\end{aligned}$.

Since $\theta$ is the angle between two vectors, its value should be between $0\; \rm radians$ and $\pi \; \rm radians$ ($0^\circ$ and $180^\circ$.) That is: $0 \le \theta < \pi$ and $0^\circ \le \theta < 180^\circ$. Apply the arccosine function (the inverse of the cosine function) to find the value of $\theta$:

$\displaystyle \cos^{-1}\left(\frac{17}{\sqrt{89}\cdot \sqrt{26}}\right) \approx 1.21 \;\rm radians \approx 69.3^\circ$ .