# MA 227 Standard S01

Points and Vectors in 3D Space

At the end of the course, each student should be able to…

• S01: 3DSpace. Plot and analyze points and vectors in three-dimensional Euclidean space.

## S01. Points and Vectors in 3D Space

• $$xyz$$ space is modeled by extending a $$z$$ axis perpendicular to the $$xy$$ plane through the origin.
• Points are given as ordered triples $$P=\<x,y,z\>$$.
• Distance/length in $$xyz$$ space may be computed using the generalized Pythagorean Theorem: $$x^2+y^2+z^2=d^2$$.
• A vector $$\vect v$$ measures a direction and magnitude in 2D or 3D space.
• Vectors are often represented by the point they point when starting from the origin: $$\vect v=\<v_x,v_y,v_z\>$$.
• Since the $$xy$$ plane is contained in $$xyz$$ space, we assume $$\<v_x,v_y\>=\<v_x,v_y,0\>$$.
• Vector addition describes the total direction and magnitude obtained by moving along the second vector after the first: $$\vect v+\vect w=\<v_x+w_x,v_y+w_y,v_z+w_z\>$$.
• Multiplication of a vector by a scalar (real number) gives the vector scaled by that real number: $$c\vect v=\<cv_x,cv_y,cv_z\>$$.
• The vector from point $$P$$ to point $$Q$$ is given by their subtraction: $$Q-P=\<q_x-p_x,q_y-p_y,q_z-p_z\>$$.
• The magnitude of a vector is given by its length: $$\|\vect v\|=\sqrt{v_x^2+v_y^2+v_z^2}$$.
• The direction of a vector is given by dividing out its length: $$\frac{1}{\|\vect v\|}\vect v$$.
• A unit vector has magnitude $$1$$.
• The standard unit vectors point in the direction of the positive coordinate axes.
• These are $$\veci=\<1,0,0\>$$, $$\vecj=\<0,1,0\>$$, and $$\veck=\<0,0,1\>$$.
• Vectors may be expressed as a linear combination of these: $$\vect v=v_x\veci+v_y\vecj+v_z\veck$$.
• The zero vector is $$\vect 0 =\<0,0,0\>$$.

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 11.1 (exercises 41-46, 61-62)
• 11.2 (exercises 1-30, 41)