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Vectors and Vector Addition
Honors/MYIB Physics
This is a vector.
It has an x-component and a y-component.
The x-component is 3 units, and the y-component is 2 units.
3 units
2 units
Here is a second (red) vector.
Its x-component is 1 unit and its y-component is 4 units.
1 unit
4 units
I can add the two vectors by drawing them head-to-tail.
The sum is called the resultant vector. It is drawn from start to end.
The resultant has an x-component of 4 units and a y-component of 6 units.
4 units
6 units
This can be found by adding the two x-components and adding the two y-components of the original vectors.
3 units 1 unit
4 units
2 units
+
+ = 4 units
6 units
One way to name the resultant vector is using components: 4, 6 .⟨ ⟩
4 units
6 units
You try it! What is the sum of the two vectors below?
The resultant vector is shown below. It has components 5, 6 .⟨ ⟩
5 units
6 units
Another way of naming a vector is by giving its magnitude and direction.
5 units
6 units
R = 5, 6⟨ ⟩
The magnitude is the length of the vector. It can be found using the Pythagorean theorem.
R = 5, 6⟨ ⟩
Rx = 5 units
Ry = 6 units
R = Rx2 + Ry
2 = 52 + 62 = 7.810 units
The direction is given as an angle. We can find it using trigonometry.
R = 5, 6⟨ ⟩
Rx = 5 units
Ry = 6 unitsR = 7.8
10 un
its
= tan−1 = 50.19°65
Our resultant vector points 7.810 units 50.19° north of east because the angle was measured from due east.
R = 5, 6⟨ ⟩
R = 7.8
10 un
its
= 50.19°
R = 7.810 units, 50.19° north of east
If I measured the angle shown below, it would be called north of west because it is measured from due west.
I could also describe this vector with a direction west of north if I measured the angle shown here.
Try it yourself! How would you name these angles using compass points?
It doesn’t matter which angle is smaller; it matters which axis you measure from!
East of North
South of East
South of West
North of West
East of South
West of South
If I know the magnitude and direction of a vector, it is easy to calculate its x- and y-components.
The vector shown below points 2.5 units 30° north of west.
A = 2.5 units, 30° north of west
30°
Its x-component is −2.5 cos 30° = −2.165 units. It is negative because it points to the left, along the −x axis.
A = 2.5 units, 30° north of west
30°
cos 30° =−Ax
AAx = −A cos 30°
Ax = −2.5 cos 30°
Its y-component is 2.5 sin 30° = 1.25 units. It is positive because it points up, along the +y axis.
A = 2.5 units, 30° north of west
30°
sin 30° =AyA
Ay = A sin 30°
Ax = −2.165 units Ay = 2.5 sin 30°
You can always find Ax = ±A cos and Ay = ± A sin if you know the vector’s magnitude and angle with the x-axis.
A = 2.5 units, 30° north of west
30°Ay = 1.25 units
A = −2.165 units, 1.25 units⟨ ⟩
Ax = −2.165 units
The components of a vector can be + or − depending on its direction, but the magnitude is always positive.
A = 2.5 units, 30° north of west
30°
A = −2.165 units, 1.25 units⟨ ⟩
The end!
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