2 Linear Combination of Vectors These two vectors are on the same line (collinear)
3 Linear Combination Since u and v are collinear: Therefore: From (2): Thus:
4 Linear CombinationDefinition: any two non-collinear vectors form a basis for the plane in which they lie, and any other vector in that plane can be written as a linear combination of these basis vectorsAssumeWe must demonstrate that these two vectors are not collinearFrom the 1st element:Since these are different, u and v are not collinear and hence form a basis for the plane.From the 2nd element:
6 Linear Combination Thus we have Note: x is coplanar with u and v Therefore:
7 Linear Combination Theorem: Are u=(2, -1, -2), v=(1, 1, 1) and w=(1, -5, -7) coplanar?Therefore the vectors are coplanar
8 Equations of Lines in the Plane In order to determine a straight line it is enough to specify either of the following sets of information:Two points on the line, orOne point on the line and its directionFor a line a fixed vector is called a direction vector for the line if it is parallel toNote: every line has an infinite number of direction vectors that can be represented as where is one direction vector for the line and t is a non-zero real number.
9 Equations of Lines in a Plane Find a direction vector for each lineThe line l1 through points A(4,-5) and B(3, -7)The line l2 with slope 4/5Any scalar multiple of (-1,-2) could also be used as a direction vector of l1a)Therefore a direction vector for l1 can be given by vector (-1, -2)b)A line with slope 4/5 that passes through the origin would pass through the point (5,4). Thus we can use direction vector (5,4) for l2
10 Equations of Lines in the Plane Pick any point P(x,y) on the line.Because P is on the line, the vector P0P (from PO to P, can be written as a scalar multiple of the direction vector d=(1,2): that is
11 Equations of Lines (2D)For each real value of the scalar t in the vector equations corresponds to a point on the line. This scalar is called the parameter for the equation of the line.d1 and d2 are called direction numbers of the line
12 UnderstandingFind vector and parametric equations of the line through points A(1,7) and B(4,0)A direction vector for this line is:Thus, a vector equation of this line is:From the vector equation we can obtain the parametric equation
13 Equations of Lines (2D)Yet another form of equation of a line evolves from solving the parametric equations for the parameter.Therefore
14 UnderstandingFor each pair of equations, determine whether or not they describe the same line
15 UnderstandingFor each pair of equations, determine whether or not they describe the same lineStep 1: compare the direction vector in both linesStep 2: see if a point on one line is also on the other lineThe direction vectors are parallelPick the point (4,4) and checkWhen r=1, we have a matchTherefore these equations are for the same line
16 UnderstandingFor each pair of equations, determine whether or not they describe the same lineStep 1: compare the direction vector in both linesStep 2: see if a point on one line is also on the other lineThe direction vectors are parallelPick the point (2,0) and checkSince the left-side does not match the right side, the lines are different
17 UnderstandingFor each pair of equations, determine whether or not they describe the same lineStep 1: compare the direction vector in both linesThe direction vectors are not parallel, therefore these lines cannot be identical
18 Equations of Lines in Space Vector Equation:Parametric Equations:Symmetric Equations:
19 UnderstandingFind the vector, parametric, and symmetric equations for the line through the points A(1, 7, -3) and B(4, 0, 2).First determine the direction vector:A vector equation is:A parametric equation is:A symmetric equation is:
20 Direction NumbersOne alternative technique for describing the direction of a line focusses on the direction angles of the line.
22 Direction Cosines In the Plane: In Space: Note: The direction cosines of a line are the components of a unit vector in the direction of the line
23 UnderstandingThe line l has direction vector (1,3,5). Find its direction cosines and thus its direction anglesSo,
24 UnderstandingDetermine the angle, to the nearest degree, that (1, 2, -3) makes with the positive x-axis.
25 Understanding Find the cosines for the line: The direction vector is: This vector is not in the upper half-space (last coordinate is not positive)So we choose:This direction vector (parallel to the first) is in the upper half-space
26 Understanding For the line we could use the direction vector However, we can obtain “nicer numbers” is we useThen a vector equation is:Parametric equations are: