Question: A strapping young man exerted a force of [10N,10N,10N] moving a box from position (1,2,3) to position (4,5,6) on a 3-D cartesian axis, where all positions are defined in meters. What is the total work done? Answer: Work is defined as follows:
W = F ⋅ d
where W is the work (scalar), F is the force vector, and d is the displacement vector. In the problem, the displacement is not exactly given to us. For that, we need to know the following relationship:
d = xf - xi
where xf is the final position and xi is the initial position. First, since we know the initial and final displacements, let’s calculate the displacement vector:
d = (4,5,6) - (1,2,3)
d = [4-1,5-2,6-3]
d = [3,3,3]
So, our displacement vector is [3,3,3]. Notice in the vector subtraction that each component of the position was subtracted individually, and all three directions combined then give the final displacement vector. With both vectors in hand, we can now calculate the work:
W = F ⋅ d
= [10N,10N,10N] ⋅ [3m,3m,3m] (1)
= 10N*3m + 10N*3m + 10N*3m (3)
W = 30J + 30J + 30J (4)
W = 90J (5)
Where J stands for Joule, which is defined to be a N*m. Notice that the work could be the exact same if say the x-component of the dot product was 10 Joules more and the z-component 10 J less. Were you able to catch the steps in there? Each equation has a number in parentheses to the far right. These are the steps of dot product calculation. Notice that step (2) is missing. In the problem, we know that the coordinates are 3-dimensional. As you grow more comfortable with dot products, step (2) becomes more of a mental note than something that needs to be officially addressed.
2-D Cross Product Question: For the following vectors c = [2,5] and d = [-4,9] calculate c x d and calculate d x c. First, let’s calculate c x d:
Step 1: Identify Two Vectors in Coordinate System as [ax,ay] & [bx,by]- These are given in the problem as [2,5] and [-4,9]. For later reference, let’s identify each component:
ax = 2
ay = 5
bx = -4
by = 9
Step 2: Multiply ax and by- Here we can easily reference step #1 and perform the following operation:
ax * by = 2 * 9 = 18
Step 3: Multiply ay and bx- Here again we can easily reference step #1 and perform the following operation:
ay * bx = 5 * -4 = -20
Step 4: Subtract step 2 from step 3, i.e. ax*by-ay*bx:
18 - -20 = 38
Step 5: Place a k-hat at the end and box your answer, you’re done! So our k-hat represents that our vector is pointing entirely in the z-direction. So our answer could be written 38*k-hat. Alternatively, we can write this
c x d = [0,0,38].
Next, let’s calculate the opposite, d x c.
Step 1: Identify Two Vectors in Coordinate System as [ax,ay] & [bx,by]- These are given in the problem as [-4,9] and [2,5]. Notice that these are OPPOSITE from last time. For later reference, let’s identify each component:
ax = -4
ay = 9
bx = 2
by = 5
Step 2: Multiply ax and by- Here we can easily reference step #1 and perform the following operation:
ax * by = -4 * 5 = -20
Step 3: Multiply ay and bx- Here again we can easily reference step #1 and perform the following operation:
ay * bx =9 * 2 = 18
Step 4: Subtract step 2 from step 3, i.e. ax*by-ay*bx:
-20 - 18 = -38
Step 5: Place a k-hat at the end and box your answer, you’re done! So our k-hat represents that our vector is pointing entirely in the z-direction. So our answer could be written -38*k-hat. Alternatively, we can write this
d x c = [0,0,-38].
What do you notice between these two different operations?
What is different?
The vector is pointing in the opposite direction (change of sign)
What is similar?
The magnitude.
The entire vector is directed in the z-direction (i.e. x and y components are 0)
What can we draw from this?
One thing that we can notice, is that
c x d = -(d x c)
Is it possible to prove this?
