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45 Cards in this Set
- Front
- Back
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Points P in the plane are represented by ordered pairs of real numbers (a1, a2); these numbers are called ________
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Cartesian Coordinates
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A directed line segment beginning at the origin with a specified magnitude and direction.
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Vector
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What are the standard basis vectors?
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i: (1,0,0)
j: (0,1,0) k:(0,0,1) |
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If the point P has coordinates (x,y,z) and P' has coordinates (x', y', z'), then the vector PP' fro m the tip of P to the tip of P' has components ________
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(x' - x, y'-y, z'-z)
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For lines in the xy plane, one simply drops the __ component
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z
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Determine the equation of the line l passing through (1,0,0) in the direction j.
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l(t) - (1,0,0) + t(0,1,0) = (1,t,0)
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Find the equations of the line in space through the point (3, -1, 2) in the direction 2i - 3j + 4k.
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x = 3 + 2t
y = -1 - 3t z = 2 + 4t |
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Find the equation of the line in the plane through the point (1, -6) in the direction of 5i - ∏j
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x = 1 + 5t
y = -6 - ∏t |
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In what direction does the line
x = -3t + 2 y = -2(t-1) z = 8t +2 point? |
v = -3i - 2j + 8k
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Do the two lines (x,y,z) = (t, -6t + 1, 2t - 8) and (x,y,z) = (3t + 1, 2t, 0) intersect?
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No
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Find the equation of the line through (2,1,-3) and (6,-1,-5).
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x = 2 + 4t
y = 1 - 2t z = -3 - 2t |
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Find the equation of the line passing through (-1,1,0) and (0,0,1).
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x = t-1
y = 1-t z=t |
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Find the equation of the line segment between (1,1,1) and (2,1,2)
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x = 1 + t
y = 1 z = 1 + t |
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The inner product of two vectors is a ______ quantity.
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scalar
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If we have two vectors a and b, and wish to determine the angle between them (the smaller angle subtended by a and b in the plane) taht they span, we would use the ______.
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inner product
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Vectors with norm 1 are called _____ vectors.
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unit
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For any nonzero vector a, a/||a|| is a ____ vector.
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unit
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When we divide vector a by ||a||, we have _______ a.
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normalized
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The length of vector a is:
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To normalize a vector a:
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The distance between vector P and Q is:
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||PQ||
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Let a and b be two vectors in R^3 and let ⊝, where 0 ≾ ⊝ ≾ pi, be the angle between them.
Then a*b = |
||a|| ||b|| cos ⊝
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Find the angle between the vectors i+j+k and i+j-k
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1.23 radians or 71 degrees
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(Cauchy-Schwarz Inequality)
For any two vectors a and b: |
|a*b| ≾ ||a|| ||b||
with equality if and only if a is a scalar multiple of b, or one of them is 0. |
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Another way of saying two vectors are perpendicular is they are ________.
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orthogonal
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If two vectors and b are orthogonal:
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a*b = 0 or cos⊝ = 0
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If v is a vector, and l is the line through the origin in the direction of vector a, then the orthogonal project of v on a is the vector ___.
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p
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The orthogonal project of v on a is the vector
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Find the orthogonal projection of i+j on i-2j
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-(1/5)(i-2j)
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A bird is flying in a straight line with velocity vector 10i+6j+k (km/hr). Suppose that (x,y) are its coordinates on the ground and z is its height above the ground.
(a) If the bird is at position (1,2,3) at a certain moment, what is its location 1 hour later? 1 minute later? (b) How many seconds does it take the bird to climb 10 meters? |
(a) (7/6, 21/10, 181/60)
(b) 36 seconds |
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Find (3i - j + k) X (i + 2j - k)
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-i + 4j + 7k
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a vector perpendicular to a and b, is
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a X b
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the area of a parallelogram spanned by a and b is
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||a x b|| = ||a|| ||b|| cos ⊝
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Find the area of the parallelogram spanned by two vectors:
a = i + 2j + 3k b = -i + k |
2√3
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Find a unit vector orthogonal to vectors i+j and j+k
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(1/√3)(i-j+k)
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The absolute value of a 3X3 determinant is:
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the volume of the parallelepiped
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Find the volume of the parallelepiped spanned by the three vectors
I+3k 2i + j - 2k 5i + 4k |
11
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Determine the equation for the plane that is perpendicular to the vector i + j + k and contains the point (1,0,0).
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x+y+z=1
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Find the equation for the plane containing the three points (1,1,1), (2,0,0), (1,1,0).
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x+y-2=0
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Two planes are called parallel when their _____ vectors are parallel.
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normal
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The distance from (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is:
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Find the distance from Q = (2,0,-1) to the plane 3x - 2y + 8z + 1 = 0.
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1/(√77)
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