Consider the following three vectors in R³: V₁ = If b = , V₂ = 1 0 , V3 = 0 is any vector in R³, explain why we can always write b as a linear combination of V₁, V2, and V3.

Q2. What will be the effect of multiplying the vector by a positive and a negative scalar. Explain in details the effect of each .case

a. u=5i−2j+3k, v=4j−2k and w=−2i−4j−k. Find ||8u - 3v + w|| b. Find the unit vector that has the same direction as the vector from the point A=(1,5) to the point B=(−1,10)

Make v3 a unit vector

Let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find the specified vector or scalar 3v - 4w.

Let v= (4,5,8) and w=(-10,-8,9) be vectors. Find the scalar component of v in the direction of w. Also, write v as a sum of two vectors, one of which is parallel to w and the other of which is perpendicular to w.

3 -i Rewrite the vector v = in the form a + ib, where vectors a and b have only real components. 4 V2

Resolve the vector a = -i+ 4j+9k into orthogonal vectors, b = i – 3j – k. one of which is parallel to