Vectors and systems of linear equations – university Year 1 maths test to download as PDF

Let \(\mathbf a=(1,2,2)\) and \(\mathbf b=(3,0,4)\).

1. Find the angle between \(\mathbf a\) and \(\mathbf b\). (2 marks)
2. Find \(\mathbf a\times\mathbf b\) and the area of the triangle with sides \(\mathbf a\), \(\mathbf b\). (2 marks)

Find the equation of the plane through \(A(1,0,0)\), \(B(0,2,0)\), \(C(0,0,3)\) and its distance from the origin.

Solve \(x+y+z=2\), \(2x+3y+z=1\), \(x-y+2z=6\).

Consider \(x+2y=3\) and \(2x+ky=c\).

1. For which \(k\) is there exactly one solution? (1 mark)
2. For \(k=4\), find the values of \(c\) giving no solution and infinitely many solutions, and describe the lines. (3 marks)

A laser beam follows \(\mathbf r=(2,1,0)+t(1,1,1)\) and hits a mirror in the plane \(x+2y+z=10\).

1. Find the point of impact. (2 marks)
2. Find the angle between the beam and the mirror. (2 marks)