What is the length of the vector
[5 cos theta, 5 sin theta]? If the vector [5,0] is rotated 36 degrees in the counter clockwise direction, what are the components of the resulting vector?
|[5 cos theta, 5 sin theta]|
= √(25cos^2Ø + 25sin^2Ø)
= √(25(cos^2Ø + sin^2Ø)
= 5 , (since cos^2Ø + sin^2Ø = 1)
Before I answer the next one, I have to know if you know about the rotation matrix .
A little, but confusing sometimes.
To find the length of the vector [5 cos theta, 5 sin theta], we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the vector [5 cos theta, 5 sin theta] forms a right triangle with the x-axis and y-axis. The x-component (5 cos theta) is one side of the triangle, and the y-component (5 sin theta) is the other side. The hypotenuse is the length we want to find.
Using the Pythagorean theorem, we have:
Length = sqrt((5 cos theta)^2 + (5 sin theta)^2)
Simplifying the equation:
Length = sqrt(25 cos^2 theta + 25 sin^2 theta)
= sqrt(25(cos^2 theta + sin^2 theta))
Since cos^2 theta + sin^2 theta = 1, we can substitute:
Length = sqrt(25 * 1)
= sqrt(25)
= 5
Therefore, the length of the vector [5 cos theta, 5 sin theta] is 5.
Now, let's consider the vector [5, 0] being rotated 36 degrees in the counter clockwise direction.
To find the new components of the vector, we can use trigonometry. Since the vector is being rotated in the counter clockwise direction, we will use the positive angle.
The x-component of the resulting vector can be found by multiplying the length of the vector by the cosine of the angle of rotation:
x-component = 5 cos(36 degrees)
Similarly, the y-component of the resulting vector can be found by multiplying the length of the vector by the sine of the angle of rotation:
y-component = 5 sin(36 degrees)
Using a calculator, we can evaluate these trigonometric functions:
x-component ≈ 5 * 0.809 = 4.045
y-component ≈ 5 * 0.588 = 2.940
Therefore, the components of the resulting vector after rotating [5, 0] by 36 degrees in the counter clockwise direction are approximately [4.045, 2.940].
To find the length of a vector [x, y], the formula is given by:
Length = sqrt(x^2 + y^2)
In this case, the vector is [5 cos theta, 5 sin theta]. So, the length of this vector would be:
Length = sqrt((5 cos theta)^2 + (5 sin theta)^2)
Simplifying the equation:
Length = sqrt(25 cos^2 theta + 25 sin^2 theta)
Length = sqrt(25 (cos^2 theta + sin^2 theta))
Now, using the trigonometric identity cos^2 theta + sin^2 theta = 1, we can further simplify the equation:
Length = sqrt(25 * 1)
Length = sqrt(25)
Length = 5
Therefore, the length of the vector [5 cos theta, 5 sin theta] is 5.
For the second part of your question, to rotate a vector [x, y] counterclockwise by angle theta, we can use the following rotation matrix:
| x' | | cos theta -sin theta | | x |
| | = | | * | |
| y' | | sin theta cos theta | | y |
Applying this to the vector [5, 0] rotated by 36 degrees, we have:
x' = 5 * cos(36) - 0 * sin(36) = 5 * cos(36)
y' = 5 * sin(36) + 0 * cos(36) = 5 * sin(36)
Therefore, the components of the resulting vector after rotating [5, 0] by 36 degrees counterclockwise are [5 * cos(36), 5 * sin(36)].