1.3 Vector addition

To add vectors, you need to add, tail-to-head with tail-to-head. Then you draw the vector from the VERY beginning to the VERY end, and dadadada – you’ve got it !

For instance, if we add 4N east with 3N south, we graphically get:

The resultant vector is in red. Note we know it’s magnitude is 5N, because it’s a right-angled triangle, and by the Pythagorean theorem (a^2 + b^2 = c^2), we thus know 4^2 + 3^2 = 25, and the sqrt(25) = 5N, which is the solution As for the angle, by SOH-CAH-TOA, we know sin(theta)=O/H=3/4, meaning theta=arcsin(3/4) (your highschool may have taught you that as sin^-1 rather than arcsin)= 48.6 degrees. In degrees true (i.e. from North clockwise in degrees), it is 48.6+90=138.6, meaning our solution is 5N 138.6 degrees True. Note that you will get marks for the correct magnitude (5), the units (Newtons, N), and the angle (138.6 degrees True).

Alternatively, to subtract vectors, you add the negative. For example, if we subtract 4N east with 3N south, we graphically get:

Compared with above, the magnitude is still the same, but the angle is no longer 138.6 degrees true. Although the angle between the green and red arrow is still 48.6 degrees, from degrees true, it is 270-48.6=221.4, meaning our solution is 5N 221.4 degrees True.

A related idea to adding/subtracting vectors is vector multiplication, of which there are various forms.

To multiply a vector and a scalar, simply multiply the magnitude of the vector by that many times.

For instance, to multiply the pink vector by 3, we get the resultant vector blue, pointing in the same direction:

As for multiplying a vector and another vector, you technically cannot. However, we do define the dot product and the cross product.

The dot product is a scalar, defined as:

For example, when working out Work, we use the dot product, i.e. W=F.s. If F=50n and s=2m, and both are in the SAME direction (i.e. theta=0), this means F.s=50*2*cos(0)=100.cos(0), and since cos(0)=1, the answer is simply 100J. Another way to think of the dot product, is components of the vector that AREN’T in the same direction of the multipled vectors are simply IGNORED. Note this answer is a scalar.

The vector product (aka cross product) is another vector, whose direction can be determined by the right-hand rule, such that if the 2nd and 3rd fingers of the right hand point in the direction of the crossing vectors, the resultant vector direction is the direction of the thumb. As for the magnitude, that is defined as:

For example, when working out Moment, we use the cross product, i.e. M=F*s. if F=50N and s=2m, and both are at 90 DEGREES to each other (i.e. theta=90), this means F.s=50*2*sin(90 degrees)=100.sin(90 degrees), and sinec sin(90 degrees)=1, the answer is simply 100Nm. Given that the moment/spin we’re producing is anticlockwise, we also need to include this. The answer is therefore 100Nm anticlockwise. Note there’s a direction, meaning the answer is a vector.

More info: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Vectors/Add2Vectors.html

http://phys23p.sl.psu.edu/phys_anim/vectors/embederQ7.101.html