Question

Given 2 vectors, split into parallel and perpendicular components

Answer 1

Answer:

In the discussion of vector addition we saw that a number of vectors acting together can be combined to give a single vector (the resultant). In much the same way a single vector can be broken down into a number of vectors which when added give that original vector. These vectors which sum to the original are called components of the original vector. The process of breaking a vector into its components is called resolving into components.

In practise it is most useful to resolve a vector into components which are at right angles to one another, usually horizontal and vertical. Think about all the problems we've solved so far. If we have vectors parallel to the x- and y-axes problems are straightforward to solve.

Any vector can be resolved into a horizontal and a vertical component. If R⃗ is a vector, then the horizontal component of R⃗ is R⃗ x and the vertical component is R⃗ y.

When resolving into components that are parallel to the x- and y-axes we are always dealing with a right-angled triangle. This means that we can use trigonometric identities to determine the magnitudes of the components (we know the directions because they are aligned with the axes).

From the triangle in the diagram above we know that

cos(θ)RxRRx=RxR=cos(θ)=Rcos(θ)

and

sinθRyRRy=RyR=sin(θ)=Rsin(θ)

Rx=Rcos(θ)

Ry=Rsin(θ)

Note that the angle is measured counter-clockwise from the positive x-axis.