**Problem1 :**What is the angle

*θ*between the vectors

*v*= (2, 5, 3) and

*w*= (1, - 2, 4) ? (Hint: your answer can be left as an expression for cos

*θ*).

*v*·

*w*= 2 - 10 + 12 = 4 . On the other hand, we know from the geometric method that

*v*·

*w*= |

*v*||

*w*| cos

*θ*. From the components we can compute |

*v*|

^{2}= 4 + 25 + 9 = 38 , and |

*w*|

^{2}= 1 + 4 + 16 = 21 . Putting all of these equations together we find that

cosθ = 4/ |

**Problem2 :**Find a vector which is perpendicular to both

*u*= (3, 0, 2) and

*v*= (1, 1, 1) .

*a*,

*b*,

*c*) such that if we dot it into either

*u*or

*v*we get zero. This gives us two equations:

3a + 2c | = | 0 | |

a + b + c | = | 0 |

Any choice of

*a*,

*b*, and

*c*which satisfies these equations works. One possible answer is the vector (2, 1, - 3) , but any scalar multiple of this vector will also be perpendicular to

*u*and

*v*.