Magnitude of the Resultant

Magnitude of the Resultant of Vectors at Right Angles

We apply the same principle to vectors that are at right angles or perpendicular to each other.

Sketching Tail-to-Head Method

The tail of the one vector is placed at the head of the other but in two dimensions the vectors may not be co-linear. The approach is to draw all the vectors, one at a time. For the first vector begin at the origin of the Cartesian plane, for the second vector draw it from the head of the first vector. The third vector should be drawn from the head of the second and so on. Each vector is drawn from the head of the vector that preceded it. The order doesn't matter as the resultant will be the same if the order is different.

Let us apply this procedure to two vectors:

\(\vec{F}_{1} = \text{2}\text{ N}\) in the positive \(y\)-direction
\(\vec{F}_{2} = \text{1.5}\text{ N}\) in the positive \(x\)-direction

We first draw a Cartesian plane with the first vector originating at the origin:

The next step is to take the second vector and draw it from the head of the first vector:

The resultant, \(\vec{R}\), is the vector connecting the tail of the first vector drawn to the head of the last vector drawn:

It is important to remember that the order in which we draw the vectors doesn't matter. If we had drawn them in the opposite order we would have the same resultant, \(\vec{R}\). We can repeat the process to demonstrate this:

We first draw a Cartesian plane with the second vector originating at the origin:

The next step is to take the other vector and draw it from the head of the vector we have already drawn:

The resultant, \(\vec{R}\), is the vector connecting the tail of the first vector drawn to the head of the last vector drawn (the vector from the start point to the end point):

Example 1: Sketching Vectors Using Tail-to-Head

Question

Sketch the resultant of the following force vectors using the tail-to-head method:

\(\vec{F}_{1} = \text{2}\text{ N}\) in the positive \(y\)-direction
\(\vec{F}_{2} = \text{1.5}\text{ N}\) in the positive \(x\)-direction
\(\vec{F}_{3} = \text{1.3}\text{ N}\) in the negative \(y\)-direction
\(\vec{F}_{4} = \text{1}\text{ N}\) in the negative \(x\)-direction

Step 1: Draw the Cartesian plane and the first vector

First draw the Cartesian plane and force, \(\vec{F}_{1}\) starting at the origin:

Step 2: Draw the second vector

Starting at the head of the first vector we draw the tail of the second vector:

Step 3: Draw the third vector

Starting at the head of the second vector we draw the tail of the third vector:

Step 4: Draw the fourth vector

Starting at the head of the third vector we draw the tail of the fourth vector:

Step 5: Draw the resultant vector

Starting at the origin draw the resultant vector to the head of the fourth vector:

Example 2: Sketching Vectors Using Tail-to-Head

Question

Sketch the resultant of the following force vectors using the tail-to-head method by first determining the resultant in the \(x\)- and \(y\)-directions:

\(\vec{F}_{1} = \text{2}\text{ N}\) in the positive \(y\)-direction
\(\vec{F}_{2} = \text{1.5}\text{ N}\) in the positive \(x\)-direction
\(\vec{F}_{3} = \text{1.3}\text{ N}\) in the negative \(y\)-direction
\(\vec{F}_{4} = \text{1}\text{ N}\) in the negative \(x\)-direction

Step 1: First determine \(\vec{R}_{x}\)

First draw the Cartesian plane with the vectors in the \(x\)-direction:

Step 2: Secondly determine \(\vec{R}_{y}\)

Next we draw the Cartesian plane with the vectors in the \(y\)-direction:

Step 3: Draw the resultant vectors, \(\vec{R}_{y}\) and \(\vec{R}_{x}\) head-to-tail

Step 4: Comparison of results

To double check, we can replot all the vectors again as we did in the previous worked example to see that the outcome is the same:

Magnitude of the Resultant