MCV4U – Grade 12 Calculus & Vectors – Cartesian Vectors Test Cartesian Vectors ● Unit Vectors: are i = [1, 0], j = [0,1] have magnitude 1 and tails at origin. ● A vector on the Cartesian plane is represented as a Cartesian vector when its endpoints are points on the 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 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 uis 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 the 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.