10.3 Components

Components of vectors

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). Inmuch 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 .

While summing a given set of vectors gives just one answer (the resultant), a single vector can be resolved into infinitely many setsof components. In the diagrams below the same black vector is resolved into different pairs of components. These components are shown as dashed lines. When added together the dashed vectors give the original black vector(i.e. the original vector is the resultant of its components).

In practice it is most useful to resolve a vector into components which are at right angles to one another, usually horizontal and vertical.

Any vector can be resolved into a horizontal and a vertical component. If $\overrightarrow{A}$ is a vector, then the horizontal component of $\overrightarrow{A}$ is ${\overrightarrow{A}}_x$ and the vertical component is ${\overrightarrow{A}}_y$ .

Block on an incline

As a further example of components let us consider a block of mass $m$ placed on a frictionless surface inclined at some angle $\theta$ to the horizontal. The block will obviously slide down the incline, butwhat causes this motion?

The forces acting on the block are its weight $mg$ and the normal force $N$ exerted by the surface on the object. These two forces are shown in the diagram below.

Now the object's weight can be resolved into components parallel and perpendicular to the inclined surface. These components are shown asdashed arrows in the diagram above and are at right angles to each other. The components have been drawn acting from the samepoint. Applying the parallelogram method, the two components of the block's weight sum to the weight vector.

To find the components in terms of the weight we can use trigonometry:

The component of the weight perpendicular to the slope $F_{g\perp }$ exactly balances the normal force $N$ exerted by the surface. The parallel component, however, $F_{g\parallel }$ is unbalanced and causes the block to slide down the slope.