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]: