Question

Find a vector vv with the same direction as uu but with a magnitude of 2, where:
u=⟨3,−3⟩.

Answer 1

The vector v with a magnitude of 2 that has the same direction as u can be found by first determining the unit vector in the direction of u and then multiplying it by 2, resulting in vector v = ⟨2√2, -2√2⟩.

Step-by-step explanation:

The direction of a vector can be maintained while scaling its magnitude by multiplying the vector by a scalar. In this case, the goal is to find a vector v that has the same direction as u = ⟡3, -3⟡, but with a magnitude of 2. To do this, first calculate the unit vector in the direction of u by dividing each of u's components by its magnitude. The magnitude of u is √(3² + (-3)²) = √18. The unit vector is then ⟡3/√18, -3/√18⟡. To obtain the desired vector v, multiply each component of the unit vector by 2, resulting in vector v = ⟡6/√18, -6/√18⟡ or, simplified, v = ⟡2√2, -2√2⟡ with a magnitude of 2.