Vector Space o Vector has a direction and a magnitude, not location o Vectors can be defined by their I, j, k components o Point (position vector) is a.

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Presentation transcript:

Vector Space o Vector has a direction and a magnitude, not location o Vectors can be defined by their I, j, k components o Point (position vector) is a location in a coordinate system o dot product of two vectors is v1.v2=|v1||v2|cos  (scalar) o cross product of two vectors is perpendicular to both vectors and its magnitude is |v1 x v2| =|v1||v2|sin  (scalar) What is v2 (red) - v1 (blue)?

Do you know how to get.. oEquation of a line passing through two points, A and B (an equation which gives the position vector of any point on the line) oEquation of a plane passing through three non-collinear points A, B and C (an equation which gives the position vector of any point on a plane) oEquation of a plane with normal vector N and passes through a point A oDistance of a point to a plane oIntersection of two planes

Line Equation oA, B are two known points on the line whose position vectors are a and b ou is a vector obtained by subtracting A and B (a and b) oAn arbitrary point P (position vector r) on the line is the sum of A (represented by position vector) and a scaled version of u (a vector) uA B P o a b r (x,y,z) = (a 1, a 2, a 3 ) + (u 1, u 2, u 3 )

Plane Equation Given points A, B, C

Projection of vector a on normal vector n is |a|cos , or is a ‘dot product’ normalised n a.n Given 3 points A, B, C on the plane, normal vector can be calculated as Plane Equation P Given a point on the plane, and the plane’s normal vector n, then the plane equation can be obtained using the fact that every vector in the plane should be perpendicular to n |n||n| n x x+n y y+n z z+d=0; where d = -n.a, n= (n x,n y,n z ) So r

Plane Equation through origin not through origin

Distance of Point A to Plane

Intersection of Planes Line equation should be of the form The intersection line should be parallel to vector Given two plane equations: So to determine the line equation we need to find a point (any point) on the line (or on both planes), e.g., let the planes intersect with the (x,y) plane (z=0) to reduce variables and solve simultaneous equations for x and y.