Week One Vectors Flashcards
What is the formula for a unit vector
1/|v| x (v)
One over the magnitude of the vector multiplied by the vector
What is the Dot product.
A • B = |A| • |B|cos(theta)
Vector A dotted with vector B = the magnitude of A x mag B x cos(the angle between them)
- The dot product tells you what amount of one vector goes in the direction of another.
What is the cross product. What is it used for ?
V x W
-To find vector perpendicular to other vectors
- Rotation of solid objects are represented by cross products
- Can be use to find the area of a parallelogram
How to calculate the area of a parallelogram using the cross product
The area a parallelogram is |U x V|
It is the magnitude of the cross product between the two vectors.
What is the normal and how do you find it.
The normal is a vector that is perpendicular to 2 other vectors. You find it by taking the cross product of the 2 vectors. |V x W|
What is a scalar triple product and what is it used for.
The triple product is A•(B x C )
It is used to find the volume of a parallel piped. = | A•(B x C ) |
What is the formula for a line between two points.
X(s) = point(1) + s(point(2) - point(1))
What is the parametric equation of the line.
X(s) = a1 + sb1
Y(s) = a2 + sb2
Z(s) = a3 + sb3
How do you know if the point is on the line.
We first assume it is
Then we use the parametric equation of the line. And sub in X , Y , Z. After that you need to find S in these equations if you get the same vale of S then point is on the line but if not then it is not on the line.
How do you know two lines intersect
You first assume they do. Then you make them equal each other and solve for S and T. If there is a contradiction then they don’t intersect
How to you find the minimum separation between two lines
You fist check if they are perpendicular by taking a dot product. Then you check if they intersect. If the don’t intersect .
Min distance = |(a2 - a1)•ṉ^|
Ṉ = b1 x b2 / ṉ^ = unit vector of Ṉ
How to find the equation of a Plane Aka (parametric vector equation).
X(s,t) = point(1) + S(point(2) - point(1)) + T(point(3) - point(1)).
What is the vector equation of the plane?
X•N = P1•N
N = B1 x B2
What is the Cartesian equation of the plane. How do you find it.
aX + bY + cZ = D
You get this by expanding the vector equation of the plane.
How do you know a line intersects the plane. If it does we’re does it intersects.
-You first need to find the Cartesian equation of the plane.(aX + bY + cZ = D)
-Then you need to find the parametric equation of the line.
-You sub the parametric equation of the line into the Cartesian equation of the plane.
- Then you solve for S
- The line usually will intersect the plane
- Sub the value of S back into the line equation. To find the point of intersection.
