1. ## vector problems

a) if position vector A has co-ordinates (2, -1) how do I calculate the unit vector in the direction OA where O is the origin? Is it 2v?

b) If I have 3 vectors P, Q, and R all with 3D co-ordinates how do write the sum t down of all 3?

c) How do I find the cosine of the angle between t and r?

Cheers

2. Originally Posted by poundedintodust

a) if position vector A has co-ordinates (2, -1) how do I calculate the unit vector in the direction OA where O is the origin? Is it 2v?

b) If I have 3 vectors P, Q, and R all with 3D co-ordinates how do write the sum t down of all 3?

c) How do I find the cosine of the angle between t and r?

Cheers
to a) A unit vector has the same direction as the given vector but only the length 1. S calculate the length of the given vector and afterwards divide the vector by it's length to get a vector with a length of 1:

$|(2, -1)| = \sqrt{2^2+(-1)^2}=\sqrt{5}$ . Therefore the unit vector has the coordinates:

$\overrightarrow{a^0} = \left( \frac{2}{\sqrt{5}}~,~ \frac{-1}{\sqrt{5}}\right)$

to b) Add the i-, j- and k-coordinates:

$(p_1, p_2, p_3) + (q_1, q_2, q_3) + (r_1, r_2, r_3) = (p_1 + q_1 + r_1 ~,~ p_2 + q_2 + r_2 ~,~ p_3+ q_3 + r_3)$

to c) Use the formula:

Let $\alpha$ denote the angle between the two vectors then:

$\cos(\alpha) = \frac{\vec r \cdot \vec t}{|\vec r| \cdot |\vec t|}$