Unit vectors Easy but tricky

[Deadawake]

Actually that's very easy question but I have some difficult to understand the logic behind .
So-"The initial velocity of an object (m/s) is Vi=1i+5j+2k. And the final velocity is Vf=3i+5j+7k. What was the change in speed of the object?"X
Solution -
|Vf|-|Vi| = √(3²+5²+7²)-√(1²+5²+2²) = 3.63 m/s
My question -
If I would do |Vf-Vi| - why is it wrong ? And what is the meaning of the scalar I get from this equation? Has to be some connection to the "change in speed". I just can't see it clearly.


Thanks again, just started to learn physics and I love it.

 

[DrClaude]

If I would do |Vf-Vi| - why is it wrong ? And what is the meaning of the scalar I get from this equation? Has to be some connection to the "change in speed". I just can't see it clearly.

That's the change in velocity, $\Delta \mathbf{v} = \mathbf{v}_f - \mathbf{v}_i$. If you divide by the time it took for that change in velocity to happen, you get the effective acceleration:
$$
\mathbf{a} = \frac{\Delta \mathbf{v}}{\Delta t}
$$
So it isn't "wrong," it just measures something else.

To visualize it, I suggest taking $\mathbf{v}_i = 1 \mathbf{i}$ and $\mathbf{v}_f = 1 \mathbf{j}$, where you have no change in speed, but an obvious change in velocity.

 

[Doc Al]

They just want the change in speed, which is the magnitude of the velocity. No deep meaning here.

And the change in speed can easily be vastly different than the change in velocity. For example, If your initial velocity was 5 m/s to the left and your final was 5 m/s to the right, your change in speed is zero but your change in velocity is 10 m/s to the right.