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Dot Products & Cross Products


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Dot Product
Vector multiplication operation in which you multiply corresponding components of two vectors and add them to products of other corresponding components.
Sqrt(V1^2+V2^2+...+Vn^2)
Length (of a vector)
Magnitude/Norm
Length of a vector
Unit Vector
Vector with length 1
Cauchy Schwarz Inequality
For any two nonzero vectors in R^n, the absolute value of their dot product is less than or equal to the product of the lengths of the vectors
Triangle Inequality
For any two nonzero vectors in R^n, the length of the sum of the two vectors is less than or equal to the sum of the lengths of each of the vectors
Law of Cosines
C^2 = A^2 + B^2 - 2ABcos@
Perpendicular
A wave in which the vibration of the medium is _______ to the direction in which the wave travels is called a transverse wave.
Orthogonal
The dot product of two vectors is 0
Normal vector (to a plane)
A vector that is perpendicular to plane in question
Right Hand Rule
Fingers - Direction of magnetic field (B)
Vector quantity
9.8m/s per second is a
vector
a displacement distinguished by magnitude and direction but not by location.
Magnitude...
... of a vector is always a positive quantity
equal vectors
have same direction and magnitude
vector sum A +B
place the initial point of vector B at the terminal point of vector A, the vector with the same initial point as A and the same terminal point as B is the sum; the…
parallelogram rul
another method to find A = B; place vectors A and B so that their initial points coincide, then complete a parallelogram that has A and B as two adjacent sides; the diagonal of…
opposite vector
a vector with the same magnitude as the original, but opposite direction.
zero vector
a vector of length zero.
scalar product
a real number(scalar) k and a vector u create the vector u create ku, which has magnitude |K| times the magnitude of u; ku has the same direction as us if k>0, and the opposite direction if k<0
position vector
a vector with its initial point at the origin; with its endpoint at (a, b) is written <a, b,> so u=<a,b>
direction angle
the positive angle between the x axis and a position vector
8
Find u · v if u = ?4,0? and v = ?2,1?.
-16
Find u · v if u = -3i + j and v = 5i - j.
56
Find u · v if u = ?8, 2? and v = ?6, 4?.
-23
Find u · v if u = 2i - 5j and v = 6i + 7j.
73
Find u · v if u = ?4, 25? and v = ?-13, 5?.
48.752
Find the angle between the following vectors if u = 5i + 7j and v = 10i + j (to the nearest thousandth).
56.575
Find the angle between the following vectors if u = ?9, 4? and v = ?2, 12? (to the nearest thousandth).
8.130
Find the angle between the following vectors if u = 3i + 5j and v = 8i + 19j (to the nearest thousandth).
99.232
Find the angle between the following vectors if u = ?20, 1? and v = ?-3, 14? (to the nearest thousandth).
129.822
Find the angle between the following vectors if u = -23i + 6j and v = 8i - 17j (to the nearest thousandth).