113k views
4 votes
Find the angle between u =the square root of 5i-8j and v =the square root of 5i+j.

1 Answer

2 votes

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.

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}}\.

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) * 1= (-8).

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


\begin{aligned} \vec{u} \cdot \vec{v} = 5 * 5 + (-8) *1 = 17 \end{aligned}.

Lengths of the two vectors

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


  • \| u \| = √(5^2 + (-8)^2) = √(89).

  • \| v \| = √(5^2 + 1^2) = √(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 \ to find the cosine of this angle:


\begin{aligned} \cos(\theta)&amp;= \frac{\vec{u} \cdot \vec{v}} v \ = (17)/(√(89)\cdot √(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((17)/(√(89)\cdot √(26))\right) \approx 1.21 \;\rm radians \approx 69.3^\circ .

User ThatSteveGuy
by
6.7k points