Grade 12 – Calculus and Vectors

Cartesian Vectors Test

**Cartesian Vectors**

**Unit Vectors:**are i = [1, 0], j = [0,1] have magnitude 1 and tails at origin.- Cartesian Vector is a representation of a vector on the Cartesian plane where the endpoints are the points on the Cartesian plane.
- If a vector, u, is translated so that the tail is at 0,0, then that vector is called a
**position vector**. Position vectors are represented in square brackets where the points are [u1 , u2]. **Magnitude of u = [u1, u2] is |u| = √(u1**^{2}**+ u2**^{2}**)**- Horizontal and vertical components of vector u can be said as [u1, 0] and [0, u1] respectively
- For vectors u = [u1, u2] and v = [v1, v2] and scalar k,
- u + v = [u1 + v1, u2 + v2]
- u – v = [u1 – v1, u2 – v2]
- kv = [kv1, kv2]
- Cartesian vector between 2 points P1(x1, y1) and P2 (x2, y2) is P1P2 = [x2 – x1, y2 – y1]
- Geometric vectors can also be written as:
- v = [ |v| cosX , |v| sinX ] where X is the angle v makes with the positive x-axis

**Dot Product**

- The dot product is defined as a∙b = |a| |b| cos X, where X is the angle between a and b.
- Dot product produces a scalar.
- For any vectors u, v, and w and scalar k,
- u and v cannot be zero are perpendicular if and only if u∙v = 0
- Commutative property
- Associative property
- Distributive property
**u**∙**u = |u|**^{2}**u ∙ 0 = 0**- If u = [u1, u2] and v = [v1, v2] then u∙v = u1*v1 + u2*v2

**Applications of Dot Product**

- For any 2 vectors u and v with angle of X between them, the projection of v on u is the vector component of v in the direction of u.
- proj
_{u}v = |v| cos X (1/|u| *|u|) or proj_{u}v = (v∙u / u∙u) u - |proj
_{u}v |= |v| cos X if 0 < X < 90 - |proj
_{u}v |= – |v| cos X if 90 < X < 180 - |proj
_{u}v |= | u∙v / |u| | - Based on the angle between u and v, you can determine what direction the projection will be.

**Vectors in 3 space**

- Same properties and formulas as Vectors in 2 space
**Orthogonal:**If 2 vectors are orthogonal, the angle between them is 90 degrees.

**Cross Product**

- The cross product between 2 vectors will find a vector that is perpendicular between both of them
- Used to find torque of it
- Direction is found by using right hand rule. Fingers point towards vector r, then bend towards vector f, thumb points in direction of the cross product
**a X b = ( |a| |b| sin X)n**where n is the unit vector orthogonal to both a and b following the right hand rule for direction and X is the angle between the vectors.- For cartesian vectors:
- If v = [v1, v2, v3] and u = [u1, u2, u3]
- u X v = [u2v3 – u3v1 , u3v1 – u1v3 , u1v2 – u2v1]
- |a X b| = |a| |b| sin X
- This also calculates the
**area of a parallelogram** - Properties
- u X v = -(v X u)
- distributive property
- associative property
- commutative property
- If u != 0 and v != 0, u X v = 0 if and only if u = mv, in other words, they must be collinear for that to happen.

**Applications of Cross Product**

- Torque is the cross product between the length of the wrench and the force applied
- Formulas for projection work in 3D and 2D
**Triple Scalar Product: a * b X c**- The cross product must be done before the dot product is done
- The volume of a parallelogram is:
**V = |w * u X v |** - Work against gravity is the dot product only with the z axis for gravity.