Answer: 5*sqrt(5)
This is the same as sqrt(125).
It approximates to roughly 11.1803
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Step-by-step explanation:
Vector u is 5 units long as indicated by the notation ||u|| = 5. Some math books would say |u| = 5, and it's the same thing. The double bars are probably more fitting because it helps emphasize we're looking for the length of a vector and we're not using absolute value bars.
The second equation ||u|| = (1/2)*||v|| can be rewritten into ||v|| = 2*||u|| telling us that vector v is twice as long compared to vector u. Since vector u is 5 units long, that must mean vector v is 10 units long.
- vector u = 5 units long
- vector v = 10 units long
The two vectors are orthogonal, i.e. they are perpendicular. The vector u+v represents the diagonal of the rectangle as shown below in the attached image. Vector u is horizontal, v is vertical, and u+v is the diagonal. Technically we don't know where vector u is pointed along a compass (it may or may not be pointing directly east), but I find it's easier to have it set up like this. The answer will be the same regardless of where vector u is pointed. Rotations don't affect vector lengths.
We can see that vector u+v effectively breaks down component-wise into the horizontal component u and vertical component v (think of them as x and y components respectively).
Furthermore, note how there are two right triangles that form when we draw in the diagonal of the rectangle.
To find the length of vector u+v, we apply the pythagorean theorem.
a^2+b^2 = c^2
( ||u|| )^2 + ( ||v|| )^2 = ( ||u+v|| )^2
( 5 )^2 + ( 10 )^2 = x^2
25 + 100 = x^2
125 = x^2
x^2 = 125
x = sqrt(125)
x = sqrt(25*5)
x = sqrt(25)*sqrt(5)
x = 5*sqrt(5) is the length of vector u+v
In other words, ||u+v|| = 5*sqrt(5)
Keep in mind that this trick using the pythagorean theorem only works when vectors u and v are orthogonal. The process to find ||u+v|| for non-orthogonal vectors u,v is a bit more complicated.