10.4 study questions - the cross product
What is the distributive property of the cross product?
(a + b) × c = a × c + a × b
What is the commutative property of the cross product?
(ca) × b = c(a × b) = a × (cb) (Where c is a scalar)
What is the cross product of two parallel vectors?
0 (zero)
What are some things that the cross product is NOT?
1. Associative: (a × b) × c ≠ a × (c × b) 2. Commutative: a × b ≠ b × a
How can the right hand rule be used to determine the direction that a × b points?
If your fingers curl in the direction from a to be (clockwise/counterclockwise), your thumb then points in the direction of a × b.
What is the geometric representation of the magnitude of cross product a × b?
Imagine that vectors a and b are two adjacent edges of a parallelogram. The area of that parallelogram is equal to |a × b|.
What is the formula for the scalar triple product?
STP = a * (b × c) (Where a, b, and c are all vectors)
Why does the STP define the volume of a parallelopiped?
The base of the parallelopiped is the parallelogram formed by b and c, so the area A of the base is A = |b × c|. The height of the parallelogram is given by the scalar component of a projected onto b × c, so h = |a| |cos θ|. So the volume of the parallelopiped V = Ah = |b × c||a||cos θ| = |a * (b × c)|
What is the geometric definition of the cross product a × b of two vectors a and b?
The vector a × b is a vector orthogonal to both a and b.
What does it mean if the triple product of three vectors is zero?
They are all coplanar
What is the geometric meaning of the STP?
Vectors a, b, and c all define a parallelopiped. (They form the three edges of one corner of a 3D parallelogram, called a parallelopiped.) The magnitude of the STP (|a * (b × c)|)defines the VOLUME of the parallelopiped.
What is the commutative property of the dot product AND cross product?
a * (b × c) = (a × b) * c
What is the cross-to-dot product property?
a × (b × c) = (a * c)b - (a * b)c
What is the transitive property of the cross product?
a × b = -b × a
What is the definition of the cross product?
a × b = <a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁>
What is the definition of torque?
torque = r × F magnitude of torque = |r × F| = |r||F| sin θ
What is the angular formula for the cross product?
|a × b| = |a||b| sin θ
