Components of Vectors, Grade 11 physics
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. While summing a given set of vectors gives just one answer (the resultant), a single vector can be resolved into infinitely many sets of 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 $\vec{A}$ is a vector, then the horizontal component of $\vec{A}$ is $\vec{A_x}$ and the vertical component is $\vec{A_y}$.

Worked Example 1: Resolving a vector into components
Question: A motorist undergoes a displacement of 250 km in a direction 30◦ north of east. Resolve this displacement into components in the directions north ($\vec{x_N}$ ) and east ($\vec{x_E}$ ).

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Step 1 : Draw a rough sketch of the original vector

Step 2 : Determine the vector component

Next we resolve the displacement into its components north and east. Since these directions are perpendicular to one another, the components form a right-angled triangle with the original displacement as its hypotenuse
Notice how the two components acting together give the original vector as their resultant.
Step 3 : Determine the lengths of the component vectors
Now we can use trigonometry to calculate the magnitudes of the components of the original displacement:

x$_{N}$ = (250)(sin 30◦) = 125 km
and
x$_{E}$ = (250)(cos 30◦) = 216,5 km
Remember x$_{N}$ and x$_{E}$ are the magnitudes of the components – they are in the directions north and east respectively.

Worked Example 2: Block on an incline plane
Question: Determine the force needed to keep a 10 kg block from sliding down a frictionless slope. The slope makes an angle of 30◦ with the horizontal.

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Step 1 : Draw a diagram of the situation
The force that will keep the block from sliding is equal to the parallel component of the weight, but its direction is up the slope.
Step 2 : Calculate F$_{g//}$
F$_{g//}$ = mg sin θ = (10)(9,8)(sin 30◦)= 49N
Step 3 : Write final answer
The force is 49 N up the slope.

Vector addition using components
Components can also be used to find the resultant of vectors. This technique can be applied to both graphical and algebraic methods of finding the resultant. The method is simple: make a rough sketch of the problem, find the horizontal and vertical components of each vector, find the sum of all horizontal components and the sum of all the vertical components and then use them to find the resultant. Consider the two vectors, $\vec{A}$ and $\vec{B}$, in Figure 11.3, together with their resultant, $\vec{R}$.

Figure 11.3: An example of two vectors being added to give a resultant​

Each vector in Figure 11.3 can be broken down into a component in the x-direction and one in the y-direction. These components are two vectors which when added give you the original vector as the resultant. This is shown in Figure 11.4 where we can see that:

In summary, addition of the x components of the two original vectors gives the x component of the resultant. The same applies to the y components. So if we just added all the components together we would get the same answer! This is another important property of vectors.

Worked Example 3: Adding Vectors Using Components
Question: If in Figure 11.4,$\vec{A}$ = 5,385 m at an angle of 21.8◦ to the horizontal and $\vec{B}$ = 5 m at an angle of 53,13◦ to the horizontal, find $\vec{R}$.

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Step 1 : Decide how to tackle the problem

Figure 11.4: Adding vectors using components.​

The first thing we must realise is that the order that we add the vectors does not
matter. Therefore, we can work through the vectors to be added in any order.
Step 2 : Resolve ~ A into components
We find the components of $\vec{A}$ by using known trigonometric ratios. First we find themagnitude of the vertical component, A$_{y}$:

Secondly we find the magnitude of the horizontal component, A$_{x}$:

The components give the sides of the right angle triangle, for which the original vector is the hypotenuse.
Step 3 : Resolve $\vec{B}$ into components
We find the components of $\vec{B}$ by using known trigonometric ratios. First we find the magnitude of the vertical component, B$_{y}$:

Secondly we find the magnitude of the horizontal component, Bx:

Step 4 : Determine the components of the resultant vector
Now we have all the components. If we add all the horizontal components then we will have the x-component of the resultant vector, $\vec{R_x}$. Similarly, we add all the vertical components then we will have the y-component of the resultant vector, $\vec{R_y}$.
R$_{x}$ = A$_{x}$ + B$_{x}$ = 5 m + 3 m = 8 m
Therefore, $\vec{R_x}$ is 8 m to the right.
R$_{y}$ = A$_{y}$ + B$_{y}$ = 2 m + 4 m = 6 m
Therefore,$\vec{R_y}$ is 6 m up.
Step 5 : Determine the magnitude and direction of the resultant vector
Now that we have the components of the resultant, we can use the Theorem of Pythagoras to determine the magnitude of the resultant, R.
R² = (R$_{x}$)² + (R$_{y}$)² = (6)² + (8)² = 100
R = 10 m

The magnitude of the resultant, R is 10 m. So all we have to do is calculate its direction. We can specify the direction as the angle the vectors makes with a known direction. To do this you only need to visualise the vector as starting at the origin of a coordinate system. We have drawn this explicitly below and the angle we will calculate is labeled α. Using our known trigonometric ratios we can calculate the value of α;

Step 6 : Quote the final answer
$\vec{R}$ is 10 m at an angle of 36,8◦ to the positive x-axis.

Exercise: Adding and Subtracting Components of Vectors
E - 1. Harold walks to school by walking 600 m Northeast and then 500 m N 40^oW. Determine his resultant displacement by means of addition of components of vectors.
E - 2. A dove flies from her nest, looking for food for her chick. She flies at a velocity of 2 m·s$^{−1}$ on a bearing of 135^o and then at a velocity of 1,2 m·s$^{−1}$ on a bearing of 230^o. Calculate her resultant velocity by adding the horizontal and vertical components of vectors.

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Physics 11.I Mathematics of Vectors

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