Answer to Question #109484 in Analytic Geometry for Caylin

Question

Consider the vectors :

u = (1,0) and v= (0,1)

1) Determine CosA, where A is the angle between u and v.

2) Determine the area of the parallelogram determined by u and v.

Answer 1

cosA = $\dfrac{u*v}{||u||*||v||}$

u * v = u₁v₁ + u₂v₂ = 1*0 + 0*1 = 0+0=0

||u|| = $\sqrt{u_1^2+u_2^2}=\sqrt{1^2+0^2}=\sqrt{1}=1$

||v|| = $\sqrt{v_1^2+v_2^2}=\sqrt{0^2+1^2}=\sqrt{1}=1$

cosA = $\dfrac{0}{1*1}=0$

Area(P) = |u * v|

|u * v| = $\begin{vmatrix} u_1 & v_1 \\ u_2 & v_2 \end{vmatrix}$

|u * v| = $\begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix}=u_1*v_2-u_2*v_1=1*1-0*0=1$

Area(P) = 1