Linear Combination of Vectors or These two vectors are on the same line (collinear)
Linear Combination Since u and v are collinear: Therefore: From (2): Thus:
Linear Combination Definition: 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 vectors We must demonstrate that these two vectors are not collinear Assume From the 1 st element: From the 2 nd element: Since these are different, u and v are not collinear and hence form a basis for the plane.
Linear Combination Note: x is coplanar with u and v Therefore: Thus we have
Linear Combination Theorem: Are u=(2, -1, -2), v=(1, 1, 1) and w=(1, -5, -7) coplanar? Therefore the vectors are coplanar
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: a)Two points on the line, or b)One point on the line and its direction For a line a fixed vector is called a direction vector for the line if it is parallel to Note: 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.
Equations of Lines in a Plane Find a direction vector for each line a)The line l 1 through points A(4,-5) and B(3, -7) b)The line l 2 with slope 4/5 Therefore a direction vector for l 1 can be given by vector (-1, -2) Any scalar multiple of (-1,-2) could also be used as a direction vector of l 1 a) 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 l 2
Equations of Lines in the Plane Pick any point P(x,y) on the line. Because P is on the line, the vector P 0 P (from P O to P, can be written as a scalar multiple of the direction vector d=(1,2): that is
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. d 1 and d 2 are called direction numbers of the line
Understanding Find 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
Equations of Lines (2D) Yet another form of equation of a line evolves from solving the parametric equations for the parameter. Therefore
Understanding For each pair of equations, determine whether or not they describe the same line
Understanding Step 1: compare the direction vector in both lines The direction vectors are parallel Step 2: see if a point on one line is also on the other line Pick the point (4,4) and check When r=1, we have a match Therefore these equations are for the same line
Understanding For each pair of equations, determine whether or not they describe the same line Step 1: compare the direction vector in both lines Step 2: see if a point on one line is also on the other line The direction vectors are parallel Pick the point (2,0) and check Since the left-side does not match the right side, the lines are different
Understanding For each pair of equations, determine whether or not they describe the same line Step 1: compare the direction vector in both lines The direction vectors are not parallel, therefore these lines cannot be identical
Equations of Lines in Space Vector Equation: Parametric Equations: Symmetric Equations:
Find 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: Understanding
Direction Numbers One alternative technique for describing the direction of a line focusses on the direction angles of the line.
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
Understanding The line l has direction vector (1,3,5). Find its direction cosines and thus its direction angles So,
Understanding Determine the angle, to the nearest degree, that (1, 2, -3) makes with the positive x-axis.
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
Understanding For the line we could use the direction vector However, we can obtain “nicer numbers” is we use Then a vector equation is: Parametric equations are: