Math 2433 Notes Week The Dot Product. The angle between two vectors is found with this formula: cosθ = a b

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1 Math 2433 Notes Week The Dot Product The angle between two vectors is found with this formula: cosθ = a b a b 3) Given, a = 4i + 4j, b = i - 2j + 3k, c = 2i + 2k Find the angle between a and c

2 Projection of a on b proj b a = ( au b) u b where au= b compba 4) Given a = 4i + 3j, b = i - 3j + 2k Find the component of a in the b direction. Find the projection of a in the b direction.

3 The angles αβγthat,, a vector makes with unit vectors i, j, and k are called direction angles of a. A unit vector with these direction angles is: cosαi+ cos βj+ cosγk 5) Find a unit vector with direction angles: α = π 6, β = π 3, γ = π 2 6) Find the direction angles of a = i - 3 k

4 7) Find all the numbers x for which the angle between c = xi + j + k and d = i + xj + k is 3 π 8) Find all numbers x such that 2i + 4j + 2xk 6i + 3j - 4xk

5 11.4 The Cross Product What operations can we use on vectors? If vectors a and b are NOT parallel, they form two adjacent sides of a parallelogram: b a a b is the vector perpendicular to this plane. So, if a b then a b = 0 The area of this parallelogram is A= a b How would we find the area of the triangle that has a and b as sides?

6 So, how do we find a b? First we need to understand determinants: a a 1 2 ab 1 2 ab 2 1 b1 b = 2 for a = ( a, a, a ) and b = (,, ) b b b, a b = Example 1) Calculate (3i + j + k) (i 2j)

7 Example 2) Calculate (i + j k) [(2i - j + k) (2i + j 2k)] 3) Find two unit vectors perpendicular to a = (1,2,-1) and b = (1,0,2)

8 4) Find the area of triangle PQR. P(2, 0, -3), Q(3, -1, 0), R(2, 3, -2) Volume of a Parallelepiped: V= (a b) c 5) Find the volume of the parallelepiped with the given edges. i - 3j + k, 3j - k, i + j - 3k How can we tell if 3 vectors are coplanar?

9 11.5 Lines Vector form: r() t = ( x0i+ y0j+ z0k) + td ( 1i+ d2j+ d3k ) Scalar form: xt () = x yt () = y zt () = z + td td td 0 3 Symmetric form: x x y y z z = = d d d Examples: 1) Find a vector parameterization for the line that passes through P(3, 2, 3) and is parallel to the line r(t) = (i + j - k) + t(2i + 3j - 3k). 2) Give the answer to #1 in scalar form 3) Give the answer to #1 in symmetric form

10 4) Give the symmetric form for the line that passes through P(2, 2, 1) and Q(2, -2, -3). 5) Find a set of scalar parametric equations for the line that passes through P(1, 2, -3) and is perpendicular to the xz-plane. Parallel lines have direction vectors that are scalar multiples of each other. 6) Give a vector parameterization for the line that passes through P(1, 2, -2) and is parallel to the line: 3( x 1) = 2( y 2) = 6( z+ 2)

11 If we have two lines, we can determine if they are parallel, coincident, skew, or intersecting. Parallel Coincident Skew Intersect 7) Determine whether the lines l1 and l2 are parallel, coincident, skew, or intersecting. If they intersect, find the point of intersection:

12 Once again, two lines (l1: r(t) = r 0 + t d and l2: R(u) = R 0 + u D) intersect if there exists numbers t and u such that r(t) = R(u) 8) Determine whether the lines l1 and l2 are parallel, coincident, skew, or intersecting. If they intersect, find the point of intersection: l1: x1() t = 1 + ty, 1() t = 1 tz, 1() t = 4+ 2t l: x( u) = 1 uy, ( u) = 1+ 3 uz, ( u) = 2u

13 If two lines are not parallel and do not intersect, then they are skew. 9) Determine whether the lines l1 and l2 are parallel, coincident, skew, or intersecting. If they intersect, find the point of intersection: l: x() t = 3+ 2, ty() t = 1+ 4, tz() t = 2 t l: x( u) = 3+ 2 uy, ( u) = 2 + uz, ( u) = 2+ 2u

14 To find the angle between two lines: cosθ = ud ud Distance from a point to a line: d( Pl,) = PP d 0 1 d 10) Find the distance from P(2, 0, 2) to the line through P0(3, -1, 1) parallel to i - 2j - 2k.

15 11.6 Planes Let N = Ai + Bj + Ck be the nonzero vector perpendicular to a plane: The equation of the plane containing Px ( 0, y0, z0) with normal N = Ai + Bj + Ck is A( x x ) + By ( y) + C( z z ) = Examples: 1) Find an equation for the plane which passes through the point P(2, 4, 0) and is perpendicular to i - 3j + 2k.

16 2) Find an equation for the plane which passes through the point P(3, -2, 3) and is parallel to the plane: 3) Find an equation for the plane which passes through the point P(1, 3, 4) and contains the line:

17 4) Find the unit normal vectors to the plane: 5) Write the equation of the plane in intercept form and find the points where it intersects the coordinate axes.

18 Angle between two planes: cosθ u u = N 1 N 2 6) Find the angle between the planes. and Ax1+ By1+ Cz1+ D Distance from a point to a plane dp ( 1, = ) A + B + C 7)Find the distance from the point P(1, -1, 2) to the plane

19 8) Find an equation in x, y, z for the plane that passes through the given points. P(1, -1, 2), Q(3, -1, 3), R(2, 0, 2)

20 Two planes will be parallel if their normal vectors are scalar multiples of each other. If two planes are not parallel, then they will intersect in a line. The line of intersection will have a direction vector equal to the cross product of their norms. 9) Find a set of scalar parametric equations for the line formed by the two intersecting planes. and