# Thread: Hey guys i have these two questions on complex numbers help please????

1. ## Hey guys i have these two questions on dot and vectors help please????

hey guys i gt tese two questions on dot and vectors , thing is im trying to revise and this topic is alien to me so hopefully by having several answers i can practice and understand....well thts the hope neways! :S

3.
(a) Define carefully the dot and vector products of two vectors a and b.
(b) Two unit vectors c and d are perpendicular. Find (c x d) . (2c + 3d).
(c) The three points. (-1, 2, 2), (2, 0, 1) and (1,2,1) form the vertices of a triangle. Use vector methods to find the angle between the two sides of the triangle which meet at (1, 2, 1). Find, also, using vector methods, the area of the triangle.

4.
(a) Define the dot and the cross product between two vectors a and b.
(b) Two unit vectors c and d are perpendicular. Find (c x d) x c.
(c) The three points (1, -2, 1), (0, 2, 1) and (-1, 1, 2) form the vertices of a triangle. Use vector methods to find the angle between the two sides of the triangle which meet at (0, 2, 1). Find, also, using vector methods, the area of the triangle.

Any help would be gratfully recieved thanks in advance

2. Originally Posted by Latkan
hey guys i gt tese two questions on dot and vectors , thing is im trying to revise and this topic is alien to me so hopefully by having several answers i can practice and understand....well thts the hope neways! :S

3.
(a) Define carefully the dot and vector products of two vectors a and b.
(b) Two unit vectors c and d are perpendicular. Find (c x d) . (2c + 3d).
(c) The three points. (-1, 2, 2), (2, 0, 1) and (1,2,1) form the vertices of a triangle. Use vector methods to find the angle between the two sides of the triangle which meet at (1, 2, 1). Find, also, using vector methods, the area of the triangle.

4.
(a) Define the dot and the cross product between two vectors a and b.
(b) Two unit vectors c and d are perpendicular. Find (c x d) x c.
(c) The three points (1, -2, 1), (0, 2, 1) and (-1, 1, 2) form the vertices of a triangle. Use vector methods to find the angle between the two sides of the triangle which meet at (0, 2, 1). Find, also, using vector methods, the area of the triangle.

Any help would be gratfully recieved thanks in advance
Assuming we're in $\displaystyle \mathbb{R}^3$ (i.e. the vectors all have three components), here are some definitions:

$\displaystyle ||u|| = \sqrt{u_1^2+u_2^2+u_3^2}$

$\displaystyle u\cdot v = \langle u_1,u_2,u_3\rangle\cdot\langle v_1,v_2,v_3\rangle = u_1v_1+u_2v_2+u_3v_3 = ||u||\,||v||\cos\theta$

From the above it is clear that if two vectors are perpendicular, their dot product will be zero.

$\displaystyle u\times v = \langle u_1,u_2,u_3\rangle\times\langle v_1,v_2,v_3\rangle = \langle u_2v_3-u_3v_2,~u_1v_3-u_3v_1,~u_1v_2-u_2v_1\rangle$

The cross-product, or vector-product as it is sometimes called, produces a vector that is perpendicular to the two vectors being crossed.

The area of a parallelogram spanned by two vectors $\displaystyle = u\times v$, and it follows that the area of a triangle spanned by two vectors $\displaystyle = \frac{1}{2}(u\times v)$, because a triangle is half a parallelogram.

I think that's all the information you need to do these problems. Post again if you need more help.

3. And just as $\displaystyle \vec{u}\cdot\vec{v}= ||\vec{u}||||\vec{v}||cos(\theta)$ so the length of $\displaystyle \vec{u}\times\vec{v}$ is $\displaystyle ||\vec{u}||||\vec{v}||sin(\theta)$, the vector itself is directed perpendicular to both, according to the right hand rule (if you curl the fingers of your right hand from $\displaystyle \vec{u}$ to $\displaystyle \vec{v}$, the vector $\displaystyle \vec{u}\times\vec{v}$ points in the direction your thumb is pointing.)