Formulas You Need To Know for Test # 3
Formulas You Need To Know for Test # 3
1. [pic]
2. [pic]
3. Standard Equation of a Sphere
The standard equation of a sphere with radius r and center [pic] is given by
[pic]
When the standard equation of a sphere is expanded and simplify, we obtain the general equation of a sphere
[pic]
4. The length (magnitude) of the 2D vector a = [pic] is given by
[pic] = [pic]
The length (magnitude) of the 3D vector a = [pic] is given by
[pic] = [pic]
5. Given a non-zero vector a, a unit vector u (vector of length one) in the same direction as the vector a can be constructed by multiplying a by the scalar quantity [pic], that is, forming
[pic]
6. The dot product of two vectors gives a scalar that is computed in the following
manner.
In 2D, if a = [pic] and b = [pic], then
Dot product = [pic][pic]
In 3D, if a = [pic] and b = [pic], then
Dot product = [pic][pic]
7. Angle Between Two Vectors: Given two vectors a and b separated by an angle [pic], [pic].
Then
[pic][pic]
Solving for [pic] gives
[pic]
8. Scalar and Vector Projection
Scalar Projection of b onto a: [pic]
Vector Projection of b onto a: [pic]
9. Determinant Formula For Cross Product
[pic]
10. Parametric and Symmetric Equations of a Line in 3D Space
The parametric equations of a line L in 3D space are given by
[pic],
where [pic] is a point passing through the line and v = < a, b, c > is a vector that the line is parallel to.
Assuming [pic], if we take each parametric equation and solve for the variable t, we obtain the equations
[pic]
Equating each of these equations gives the symmetric equations of a line.
[pic]
11. Standard and General Equations of a Plane in the 3D space
The standard equation of a plane in 3D space has the form
[pic]
where [pic] is a point on the plane and n = < a, b, c > is a vector normal (orthogonal to the plane). If this equation is expanded, we obtain the general equation of a plane of the form
[pic]
12. Angle Between Two Planes
Let [pic] and [pic] be normal vectors to these planes. Then
[pic]
Solving for [pic] gives
[pic]
13. Distance between a point Q not on a plane to the plane.
Let P be a point on the plane and let n be a normal vector to the plane. Then
[pic]
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