Vector questions.....................…?
step by step method would be appreciated
4. Given that
u = 7 i − 8 j and v = −3 i − 6 j − 5 k
4.1 Find a vector in the same direction as u but equal in magnitude to v.
4.2 Find a vector in the same direction as v but with a magnitude of 13.
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4.1 The magnitude of v, |v|, is √((-3)^2 + (-6)^2 + (-5)^2) = √(9 + 36 + 25) = √70.
The magnitude of u, |u|, is √((7^2 + (-8)^2) = √(49 + 64) = √113.
The vector requested is (7i - 8j) • √70/√113
4.2 The vector requested is (-3i - 6j - 5k) • 13/√70
For any nonzero vector u, a unit vector in the direction of u is obtained by dividing u by its magnitude ||u||. Then, to get a vector in that direction with a specified magnitude, multiply the unit vector by the desired magnitude.
4.1 ||u|| = â(7² + (-8)²) = â(105)
||v|| = â((-3)² + (-6)² + (-5)²) = â(70).
A unit vector in the direction of u is
u/||u|| = (7/â(105)) i - (8/â(105)) j
So a vector in the direction of u with the magnitude of v is â(70) times this.
â(70) u/||u|| = 7â(70)/â(105) i - 8â(70)/â(105) j
4.2 Play a similar game. For a unit vector in the direction of v by dividing each component by â(70). Then, multiply that by 13.