3D Coordinates
3D Pythagoras
\[a^2+b^2=c^2$$
$$\to$$
$$a^2+b^2+c^2=d^2\]
Distance between two points in 3D
\[d=\sqrt{ (\Delta x)^2+(\Delta y)^2+(\Delta z)^2 }\]
Find the distance from the origin to the point \(\mathbf{P}(7,7,7)\)
\(7^2+7^2+7^2=147\)
\(\text{Distance}=\sqrt{ 147 }\)
The coordinates of \(A\) and \(B\) are \((5,3,-8)\) and \((1,k,-3)\) respectively. Given that the distance from \(A\) to \(B\) is \(2\sqrt{ 10 }\), find the possible values of \(k\)
- \((3\sqrt{ 10 })^2=(1-5)^2+(k-3)^2+(-3--8)^2\)
- \(90=16+(k-3)^2+25\)
- \(49=k^2-6k+9\)
- \(k^2-5k-40=0\)
- \(k=10\), \(k=-4\)
Angle between a vector and the x-axis
\(\(\cos\theta _{x}=\frac{x}{|\mathbf{a}|}\)\)
- Where \(\mathbf{a}\) is the vector and \(x\) is the x coordinate of the vector