Below is a diagram of the dotted cube with edges of length 4
my attempt:
t=-u+v=(-4,4,0)
t-2r=(-4,4,0)-2(-4,4,4)=(4,-4,-8)
u+v+w+r+s+t=(4,0,0)+(0,4,0)+(0,0,4)+(-4,4,4)+(4,-4,4)+(-4,4,4)=(0,8,16)
is i am right can any one give me right direction
[Math] Below is a diagram of the dotted cube with edges of length 4
vectors
Related Solutions
Maybe one way that will help the understanding is to consider the integral $$\int_0^{\theta_0} {\hat \theta} d{\theta}$$
I am assuming $\hat \theta$ defined in this way: $${\hat \theta} = \begin{bmatrix}-\sin(\theta)\\\cos(\theta)\end{bmatrix}$$
It will be unit vector because of the trigonometric identity: $\sin(\theta)^2+\cos(\theta)^2 = 1$. We can also convince ourselves that it will be orthogonal to $\begin{bmatrix}r\cos(\theta)\\r\sin(\theta)\end{bmatrix}$ which is often defined to be the radial component.
If we approximate the integral using a computer, we see a relation between the angle integrated, the unit circle and the line integral result:
The end points of the blue line are angles $0$ and $\theta_0$ on the unit circle. And the red vector is the vector from point $(1,0)$ pointing onto the unit circle at angle $\theta_0$. (I could not find any arrowhead to indicate the direction).
Two examples: $p_1=(0,0),q_1=(1,1),\ \ p_2=(2,0),q_2=(0,1)$ $p_1=(0,0),q_1=(1,1),\ \ p_2=(2,1),q_2=(0,-1)$ The first example will have positive values for both s's. The second example will have a negative value for $s_2$.
I'll let you do the arithmetic.
Best Answer
$$t-2r=(-4,4,0)-2(4,4,4)=(-12,-4,-8)$$
$$u+v+w+r+s+t=u+v+w+u+v+w+u+w+v-u=$$ $$=2u+3v+3w=(8,12,12)$$