## triple x and y components?

Carmenwalf
Posts: 1
Joined: Mon Aug 20, 2018 5:36 pm

### triple x and y components?

a vector is 12 m and at 45 degrees. If the x and y components are tripled,what is the magnitude of th new vector?

I tried solving with
Vx=12cos(45)
Vy=12sin(45)
and triple those. Is that correct? if so whats next?

jeff
Posts: 38
Joined: Mon Mar 05, 2018 11:54 pm

### Re: triple x and y components?

Hi Carmenwalf,
Carmenwalf wrote:
Tue Aug 21, 2018 5:50 pm
a vector is 12 m and at 45 degrees. If the x and y components are tripled,what is the magnitude of th new vector?

I tried solving with
Vx=12cos(45)
Vy=12sin(45)
and triple those. Is that correct? if so whats next?
Yes, that will work! So once you have found the components and then tripled them, you can plug them into the formula for the magnitude of a vector:

(If the image above does not show, the text form is: |V| = sqrt( Vx^2 + Vy^2) )

So plug in your new V_x and V_y into that formula to get the answer. What do you get for that?

GloriaVipsy
Posts: 1
Joined: Mon Sep 17, 2018 1:38 pm

### Re: triple x and y components?

Ihad the same problem with different numbers. magnitude = 5 and angle is 30. I started by calculating the components, and tripled them. I then used the Pythagorean theorem. I got an answer of 15. Will it alwys just triple the total magntidue?

jeff
Posts: 38
Joined: Mon Mar 05, 2018 11:54 pm

### Re: triple x and y components?

Hi GloriaVipsy ,
GloriaVipsy wrote:
Mon Sep 17, 2018 2:01 pm
Ihad the same problem with different numbers. magnitude = 5 and angle is 30. I started by calculating the components, and tripled them. I then used the Pythagorean theorem. I got an answer of 15. Will it alwys just triple the total magntidue?
Yes, that's right! If you multiply the components of a vector by the same positive number, then the total magnitude will be multiplied by that number.

To see this, let's say we have a vector with magnitude V, and components V_x and V_y, so that:

$\large&space;V_{\rm&space;old}&space;=\sqrt{V_x^2+V_y^2}$

The new magnitude is what we get when we multiply each of those components by 3:

$\large&space;V_{\rm&space;new}&space;=\sqrt{(3V_x)^2+(3V_y)^2}$

But we can pull the 3 out of the parenthesis, and then out of the square root:

$\large&space;V_{\rm&space;new}&space;=\sqrt{3^2(V_x^2+V_y^2)}\\&space;\phantom{&space;}&space;\quad&space;V_{\rm&space;new}=3\sqrt{&space;V_x^2+V_y&space;^2}$

and what is now under the square root sign, is our original magnitude!

$\large&space;V_{\rm&space;new}=3V_{\rm&space;old}$

So this shows that multiplying each component by 3, will triple the overall magnitude. If you change the "3" in the above equations to a positive variable a, you see it works more generally.

You can explore this further. What happens if you multiply both by a negative number? or how about multiplying one component by -3 and the other by +3? or how about a 3-dimensional vector with components V_x, V_y, and V_z?