Vectors in 3D space
When we deal with vectors in space, we have one more dimension. In the coordinate system, the z-axis which is perpendicular to the xy plane is added. The function and principle of vectors remain exactly the same in three dimensional space as it is within a two dimensional plane.
In the coordinate system below, the x-axis is red, the y-axis is green, and the z-axis is blue. The z-axis base vector is denoted by the letter k.
The vector shown in the figure, which has a position vector at a point of (-10,3,2), is
A vector in space can be thought of as a space diagonal of a rectangular prism. The above could be thought of as a triangle with edge lengths of 10, 3 and 2. In this case, the length of the space diagonal is
So the length is
In general the length of a 3D vector is
Vector a is
Example 1
Point A(1,3,2) and point B(3,2,6) are in XYZ space. Find the vector AB.
Draft the situation. The draft does not have to be drawn in the coordinate system.
The position vectors of A and B are
In this case, vector AB is obtained by going against the position vector of point A and along the position vector of point B.
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