#$&*
course Mth 277
9/15/13
If your solution to stated problem does not match the given solution, you should self-critique per instructions at
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
qa 09_3
The dot product of two vectors is equal to the sum of the products of their components: •If A = a_1 i + a_2 j + a_3 k andB = b_1 i + b_2 j + b_3 k, then A dot B = a_1 b_1 + a_2 b_2 + a_3 b_3.
The dot product is also equal to || A || * || B || cos(theta), where theta is the angle between the vectors. So
•
A dot B = a_1 b_1 + a_2 b_2 + a_3 b_3
•
A dot B = || A || * || B || cos(theta).
So for example if we know the components of A and B, we can easily find the dot product and the magnitudes of the two vectors. Having found the magnitudes and the dot product we can use the second relationship to get
cos(theta) = A dot B / || A || || B || so that
theta = arcCos( A dot B / || A || || B || ).
Two vectors are perpendicular to one another if the angle between them is 90 degrees. The cosine of 90 degrees is zero, and if the cosine of an angle between 0 and 180 degrees is zero the angle is 90 degrees. So two vectors are perpendicular if, and only if, their dot product is zero.
If the dot product of two vectors is zero we say that the vectors are orthogonal. In two or three dimensions, that means that the angle between the two vectors is 90 degrees.
Section 9.3
If you have a good Precalculus II background (or equivalent) with an appropriate introduction to vectors, or if you have worked through the recommended qa's on vectors, from Precalculus I, you should have little trouble with the query questions. So no q_a_ questions are included with this assignment.
For your convenience here's a listing of the Query questions:
1) 9.3.4 Find v dot u when v =<1,-5,0> u =<0,-4,2>.
solution:20
2) 9.3.6 Find v dot w when v = 4i + j and w =3i + 2k.
solution: -12
3) 9.3.10 Determine whether v = 5i - 5j + 5k and w = 8i - 10j -2k are orthogonal.
soltuion: no since the dot product =80
4) 9.3.12 Let v = 4i - 2j + k and w = -2i + j - k. Evaluate (v dot w) * w.
soluiton:22i,-11j,11k
5) 9.3.16 Find the angle between v = 2i +3 k and w = -j + 4k.
theta=cos^-1(12/||V||||W||)
6) 9.3.20 Find the scalar and vector projections of v = i - 2j onto w = j - 2k.
7) 9.3.24 Find two distinct unit vectors orthogonal to both v = i + 2j -2k and w = i + j - 2k.
8) 9.3.30 Find x so that v = 2i - xj + 3k and w = -2i + j + xk are orthogonal.
9) 9.3.32 Give the direction cosines and direction angles of v = i - 4j.
10) 9.3.36(b,c,d) Let v = i - j + 4k and w = -i + 3j + 2k. Find cos(theta). Find s such that v is orthogonal to sv - w. Also find t such that v - tw is orthogonal to w
11) 9.3.40 Find the work performed when a force F = (6/11)i - (2/11)j + (6/11)k is applied to an object moving along the line from P(3,5,-4) to Q(-4,-9,-11).
"
@&
Not a bad start, but you need to submit the Query for this assignment, and the QA is strongly recommended.
*@