Math ch.11
Find k so that the given vectors are parallel and perpendicular
(5,8)(-2,-k) a)parallel:(-2,-k) = j(5,8) 5j = -2 j = -2/5 8j = -k k = -8j k = 16/5 B) perpendicular 5(-2) + 8(-k) = 0
Classify triangle ABC as isosceles, equilateral, or scalene
***DON'T GRAPH BC U COULD DO IT WRONG Instead find the distance between the 3 points with distance formula A(2,0,1) B(-1,5,0) C(-1,0,-3) Find AB, BC, and AC to compare
Theorem 5
1. u•v = v•u (commutative) 2. r(u•v) = (ru) • v = u•rv (associative) 3. u(v+w) = uv + uw (distributive) 4. |u•v| <= ||u|| ||v|| (Cauchy-Schwartz inequality) 5. v•v = ||v||^2 (norm property)
Resolve each vector with given magnitude and making the given angle with the horizontal into its horizontal and vertical.
13, 45 degrees (x,y)(cos theta, sin theta) (13cos45)i + (13sin45)j (13cos45,13sin45)
Orthogonal
2 vectors that have an angle between them that is 90 or they multiply to equal zero
A force F with given magnitude and direction with respect to the horizontal is applied to an object. Resolve F into its horizontal and vertical components.
35N, 45 degrees (35cos45)i + (35sin45)j
Find a set of parametric equations of each line.
Containing (4,7) and (0,6) Take v = (4,7)-(0,6) = (4,1) which is the m1,m2) Now using (0,6) and (4,1) x = 0 + 4t y = 6 + t
Whenever it asks for dot product and they don't define u and v, then
Define u and v Ex) (-3,5,6)(1,1,3) Let u = (-3,5,6) Let v= (1,1,3)
A downstream current and an across-stream wind current at 5km/h give a sailboat an effective speed of 13km/h. What is the speed of the downstream current and the angle between the wind current and the path of the sailboat?
Draw a triangle with 5 and d as legs and 13 as hypotenuse. Let d = downstream current = ||d|| = square root of 13^2-5^2 = 12 Let theta equal the angle headed Cos theta = 5/13 Theta = 67.4 degrees
A plane's heading is 160 degrees and it's air speed is 350mph. If the west wind is blowing at 20mph, what are the plane's ground speed and true course.
Draw graph Use law of cosines to find ground speed. ||v||^2 = 350^2 + 20^2 - 2(350)(20)cos110 v = 357 Use law of sines to find the true course angle. sin alpha/20 = sin 110/357 Alpha = 3 degrees. True course: 160 degrees - 3 degrees = 157 degrees Make conclusion: the ground speed is 357mph and true course is 157 degrees
Example of work
Force 15N at 24 degrees up a vertical cable a distance of 25m. Draw a triangle with 25m vertical, force as hypotenuse and angle 24 between base and hypotenuse. Since the force has to be in the direction of the vertical cable, theta is equal to 90-24 to get the complementary angle of 66 degrees.
Use the Pythagorean theorem to determine whether this triangle ABC is a right triangle or not
Given: A(10,5,1) B(-2,2,5) C(-2,5,1) First find the distances : AB,BC,AC Then determine which is the hypotenuse(biggest) Then use Pythagorean theorem BC^2 + AC^2 = AB^2 Show the distance calculations and combine BC and AC by adding after to get (144,9,16)=(144,9,16) or use calculator to find exact value and show that it is a right triangle *** could also not be a right triangle
Theorem 7
If u and v are nonzero orthogonal vectors in the plane and t is any vector in the plane, then t = (u•t/||u||^2)u + (v•t/||v||^2)v = (u•t/u•u)u + (v•t/v•v)v
Theorem 6
If u and v are not parallel vectors in the plane, then every vector in the plane can be written as a unique linear combination of u & v
If it says the wind is blowing from 40 degrees...
If u draw the graph 40 degrees would be to the right so if the wind is blowing from there then the arrow is to the left of the initial air speed
Cauchy-Schwarz Inequality
If x and y are vectors in Rn, then |x•y| <= ||x|| ||y||
When it says from 40 degrees then
In the graph there will be a 40 degrees from where the arrow is
Theorem 3
Let u and v be vectors in a vector space v and let r be a scalar. 1. ||v|| >= 0 2. ||v|| = 0 if and only if v = 0 3. ||rv|| = ||r|| • ||v|| 4. ||u+v|| <= ||u|| + ||v||
Find a vector with norm 10 and parallel to v(8,-3) but having a different opposite direction
Let v' = kv 10 = |k| ||v|| k = -10/root 73 bc opposite direction v' = -10/root 73 (8,-3)
Show that the set (3,0)(0,-1) is a basis for the set of all vectors in the plane.
Let v=(x,y) There are unique a and b such that v = a(3,0) + b(0,-1) (x,y) = (3a,-b) a = x/3 b = -y Thus (3,0)(0,-1) is a basis for the set of all vectors in the plane.
Let t = (4,-3). write t as a sum of components in the direction of u and vector of v orthogonal to u.
Let v=(x,y) x+5y = 0 x=-5 and y=1 V(-5,1) Use the t = (u•t/||u||^2)u + ... t=-11/26u - 23/26v
Find a set of parametric equations of the line containing the given point P and parallel to the given vector v.
P(-3,2); v(6,-5) (x,y) = (-3,2) + t(6,-5) x = -3+6t y = 2+(-5t)
Find the coordinates of 2 points on the x-axis that are 7 units from (4,6,3)
Since it says x-axis, write "Let the point (x,0,0) Do distance formula for (4,6,3) and (x,0,0) = 7 x^2 - 8x + 12 = 0 x = 6 or x=2 Therefore (6,0,0) or (2,0,0) *** if it says y-axis or z-axis make sure u do (0,y,0) or (0,0,z)
Norm of a vector
Square root of the 2 coordinates squared Ex) u = (3,-4) ||u|| = square root of (3)^2 + (-4)^2
Bearing
The angle theta the velocity makes with due north, 0 degrees <= theta < 360 degrees
Dot product
U • V = (9,6) (-6,4) = 9(-6) + 6(4)
Find the vector components of t in the direction of u and in the direction of v.
Use t = (u•t/||u||^2)u + (v•t/||v||^2)v Ex) t=(5,-2) u=(4,1) v=(-2,8) t=(5(4) + (-2)(1))/(square root of 4^2 + 1^2)^2(u) + (-2(5) + 8(-2))/(root of (-2)^2 + 8^2)^2(v) Leave as u and v don't plug in u and v values Therefore t=18/17u - 13/34v
Let P(1,-2,-3) and Q(1,3,5)
Use the distance formula for three space to find PQ = root 89
Find the work done by the force F with given magnitude and direction in moving an object the given distance at the given angle. Force 91N at 10 degrees along a ramp 100m long at 10 degrees.
W = Fdcos theta W = 91N(100m)(cos 10-10) W = 9100cos0 W = 9100J
3-D space
X-axis: diagonal Y-axis: horizontal Z-axis: vertical
Write t=(-9,1) as a linear combination of u=(-1,5) and v=(3,7)
t = au+bv (-9,1) = (-1a,5a) + (3b,7b) -1a + 3b = -9 5a + 7b = 1 b = -2, a=3 t=3u-2v
Write the given vector t as a sum of components parallel and perpendicular to the unit vector with given direction.
t: magnitude 13.5 direction 120 degrees, 30 degrees u = (cos 30, sin 30) V=(cos 120, sin 120) Make sure angles are 90 degrees apart t = 0u + 13.5v(cos 90 + sin 90) Look at hw examples as well***
Write the given vector t as a sum of components parallel and perpendicular to the unit vector with the given direction.
t: magnitude 20, direction:80 degrees, 45 degrees Draw a graph with the two degrees as vectors with u being 45 and v being perpendicular to 45 (135). Draw vector t as a dashed lines parallel to vector v(draw on top of) (Take 45 + 90) Take 80 degrees - 45 degrees = 35 u = (cos45,sin45)= (root2/2, root2/2) v = (cos135, sin 135) = (-root2/2,root2/2) t = magnitude(cos difference of original angles)(u) + magnitude(sin difference of original angles)(v) t = 20 cos 35(root2/2, root2/2) + 20sin35(-root2/2, root2/2)
Find the measure of the angle between the given pairs of vectors to the nearest degree.
u•v = ||u|| ||v|| cos theta Solve for cos theta
Triangle Inequality
||u + v|| <= ||u|| + ||v||