Question

In a 2-dimensional Cartesian coordinate system the y-component of a given vector is equal to that vector's magnitude multiplied by which trigonometric function, with respect to the angle between vector and y-axis?

a. sine
b. cosine
c. tangent
d. cotangent

Answer 1

(b) cosine

Step-by-step explanation:

In a 2-dimensional Cartesian coordinate system, a vector has a x-component and/or a y-component. To get these components, the magnitude of the vector is resolved with respect to the x-axis and the y-axis by multiplying it (the magnitude) by some trigonometric function with respect to the angle between the vector and the x or y axis.

For example, given a vector A of magnitude A which makes an angle α with the x-axis and an angle β with the y-axis, the x and y components of the vector A can be found as follows;

i. x-component is given by
`A_(x)`

`A_(x)` = A cos α (with respect to the angle between A and the x-axis) or

`A_(x)` = A sin β (with respect to the angle between A and the y-axis)

ii. y-component is given by
`A_(y)`

`A_(y)` = A sin α (with respect to the angle between A and the x-axis) or

`A_(y)` = A cos β (with respect to the angle between A and the y-axis)

Therefore, the y-component of a vector in a 2-dimensional Cartesian coordinate is given by the product of the magnitude of the vector and the cosine of the angle between the vector and the y-axis.

Answer 2

Option b, cosine.

Step-by-step explanation:

Below you can see an image that illustrates this situation.

Remember that for a triangle rectangle with a given angle θ, we have:

Cos(θ) = (adjacent cathetus)/(hypotenuse)

In the image, you can see a vector of magnitude M, and the angle θ defined between the vector and the positive y-axis.

In this case, the y-component is the adjacent cathetus and the hypotenuse is the magnitude of the vector.

Then we will have:

Cos(θ) = (adjacent cathetus)/(hypotenuse) = y/M

solving that for y, we get:

y = Cos(θ)*M

Then the y-component is the vector's magnitude multiplied by the cosine of the angle between the vector and the y-axis.

The correct option is b.