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Multiplication with Vectors. Scalar Multiplication Dot Product Cross Product. Objectives . TSW use the dot product to fin the relationship between two vectors. TSWBAT determine if two vectors are perpendicular. A bit of review. A vector is a _________________
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Multiplication with Vectors Scalar Multiplication Dot Product Cross Product
Objectives • TSW use the dot product to fin the relationship between two vectors. • TSWBAT determine if two vectors are perpendicular
A bit of review • A vector is a _________________ • The sum of two or more vectors is called the ___________________ • The length of a vector is the _____________
Find the sum • Vector a = < 3, 9 > and vector b = < -1, 6 > • Find • What is the magnitude of the resultant. • Hint* remember use the distance formula.
Multiplication with Vectors Scalar Multiplication Dot Product Cross Product
Scalar Multiplication: returns a vector answer Distributive Property:
Multiplication with Vectors Scalar Multiplication Dot Product Cross Product
Dot Product • Given and are two vectors, • The Dot Product ( inner product )of and is defined as • A scalar quantity
Finding the angle between two Vectors a a - b θ b
Example • Find the angle between the vectors:
Classify the angle between two vectors: Acute : ______________________________________________ Obtuse: _____________________________________________ Right: (Perpendicular , Orthogonal) _______________________
example THEOREM: Two vectors are perpendicular iff their Dot (inner) product is zero. • Given three vectors determine if any pair is perpendicular
Ex 3: Find the unit vector in the same direction as v = 2i-3j-6k
Ex 5: (c) 3v (d) 2v – 3w (e)
Ex 6: Find the angle between u = 2i -3j + 6k and v = 2i + 5j - k
Ex 7: Find the direction angles of v = -3i + 2j - 6k
Any nonzero vector v in space can be written in terms of its magnitude and direction cosines as: Ex 9: Find the direction angles of the vector below. Write the answer in the form of an equation. v = 3i – 5j + 2k
We can also find the Dot Product of two vectors in 3-d space. • Two vectors in space are perpendicular iff their inner product is zero.
Example • Find the Dot Product of vector v and w. • Classify the angle between the vectors.
Projection of Vector a onto Vector b a b a b Written :
Example: Find the projection of vector a onto vector b :
Decompose a vector into orthogonal components… Find the projection of a onto b Subtract the projection from a The projection, and a - b are orthogonal a b a-b
Multiplication with Vectors Scalar Multiplication Dot Product Cross Product
