|u| = 3, |v| = 2, and the angle between u and v (when placed tail to tail) is 45 degrees. Find |2u| + |3v|
Make a sketch and you will see that
if we let |2u| + |3v| = R
R^2 = 6^2 + 6^2 - 2(6)(6)cos135°
= 72 - 72(-√2/2)
= 72 + 36√2
R = √(72 + 36√2 ) = appr 11.09
Well, since |u| = 3 and |v| = 2, let's calculate |2u| and |3v| separately.
|2u| = 2 * |u| = 2 * 3 = 6
|3v| = 3 * |v| = 3 * 2 = 6
Now, let's add these values together:
|2u| + |3v| = 6 + 6 = 12
So, |2u| + |3v| = 12. Voilà!
To find |2u| + |3v|, we need to calculate the magnitude (length) of 2u and 3v, and then add them together.
Step 1: Calculate 2u
Since |u| = 3, multiplying it by 2 gives us 2u = 2 * 3 = 6.
Step 2: Calculate 3v
Since |v| = 2, multiplying it by 3 gives us 3v = 3 * 2 = 6.
Step 3: Calculate |2u| + |3v|
Adding the magnitudes, |2u| + |3v| = |6| + |6| = 6 + 6 = 12.
Therefore, |2u| + |3v| = 12.
To find |2u| + |3v|, we need to find the magnitudes of 2u and 3v first.
Given that |u| = 3, we can calculate |2u| by multiplying the magnitude of u by 2:
|2u| = 2 * |u| = 2 * 3 = 6.
Similarly, given that |v| = 2, we can calculate |3v| by multiplying the magnitude of v by 3:
|3v| = 3 * |v| = 3 * 2 = 6.
Now we need to find the angle between 2u and 3v. Since the angle between u and v is given as 45 degrees, we know that the angle between 2u and 3v is also 45 degrees because we are simply scaling the vectors.
Finally, we can calculate |2u| + |3v| by adding the magnitudes:
|2u| + |3v| = 6 + 6 = 12.
Therefore, |2u| + |3v| is equal to 12.