Grade 12 – Calculus and Vectors
Cartesian Vectors Test
- 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| = √(u12 + u22)
- 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
- 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.
- 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
- 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.