Probability, Matrix, and Linear Algebra

Question 1. Find x and y so that [2x − y, x + 3y] = [1, 25].

Question 2. Write the vector [3, 5] as a linear combination of [6, 4] and [3, 1].

Question 3. Find the angle between the vectors [1, 2, 2] and [3, 4, 5].

Question 4. (a) Find the cross product [2, 3, 4] × [7, 6, 5]. (b) Find a unit vector that is orthogonal to both [2, 3, 4] and [7, 6, 5]. (c) Compute the dot product of the unit vector that you found in Part (b) with each of [2, 3, 4] and [7, 6, 5].

Question 5. Find the shortest vector among the following 3 vectors:

[3, 0, −5], [−1, 2, −4], [2, 3, 2].

Question 6. Find the equation of the line that passes through the points (1, 3, 4) and (5, 8, 6).

Question 7. Find the equation of the plane that passes through the point (1, 3, 4) and is normal to [4, 3, 2].