Find the vector v with the given magnitude and the same direction as u.
Magnitude: ||v|| = 8
Direction: u = ‹6, -6›
STEP 1: Find the magnitude of u.
STEP 2: Use your result from Step 1 to compute a unit vector, u1, in the same direction as u.
u1=<?/√ (?) , ?/√ (?)
STEP 3: Multiply u1 by the given magnitude to find v, a vector with the same direction as u and magnitude 8.
v=<?,?>
For step 1, my answer is √72, can someone please help me on step 2 and step 3 please!!
STEP 1: The magnitude of vector u can be found using the formula:
||u|| = √(u1^2 + u2^2)
Plugging in the values given:
||u|| = √(6^2 + (-6)^2)
= √(36 + 36)
= √72
= 6√2
Therefore, the magnitude of vector u is 6√2.
STEP 2: To find the unit vector u1 in the same direction as u, divide each component of u by its magnitude:
u1 = (u1/||u|| , u2/||u||)
Plugging in the values:
u1 = (6/6√2 , -6/6√2)
= (1/√2 , -1/√2)
Therefore, the unit vector u1 in the same direction as u is (1/√2 , -1/√2).
STEP 3: Multiply the unit vector u1 by the given magnitude to find vector v:
v = (1/√2 , -1/√2) * 8
= (8/√2 , -8/√2)
= (4√2 , -4√2)
Therefore, the vector v with the same direction as u and magnitude 8 is (4√2 , -4√2).
Sure! Let's go through step 2 and step 3.
Step 2: To find a unit vector u1 in the same direction as u, divide each component of u by its magnitude (which you found to be √72 in step 1).
So, u1 = (6/√72, -6/√72).
Step 3: To find the vector v with the given magnitude (8) and the same direction as u, multiply u1 by the magnitude.
v = 8 * u1 = (8 * (6/√72), 8 * (-6/√72)).
Simplifying this, we have:
v = (48/√72, -48/√72).
So, the vector v with the given magnitude (8) and the same direction as u is v = (48/√72, -48/√72).
Note: If you'd like, you can simplify the expression further by rationalizing the denominator (√72).
u is (6,-6) so u1=1@-45= .701,-.701 ( or 1/sqrt2 if you wish).
so to get to magnitude 8, then
v=8@-45= 8/sqrt2,-8/sqrt2)