MA19.PRE.11

Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.11
Find the scalar (dot) product of two vectors as the sum of the products of corresponding components and explain its relationship to the cosine of the angle formed by two vectors.

Unpacked Content

Knowledge

Students know:

The formula and alternative formula for dot product.
The properties of the dot product.
The formula for the angle between two vectors.
The relationship between the dot product and orthogonal vectors.
Projection of a vector onto another vector.
Vector components of v.

Skills

Students are able to:

Find the dot product of two vectors.
Find the angle between two vectors.
Use the dot product to determine if two vectors are orthogonal.
Find the projection of a vector onto another vector.
Express a vector as the sum of two orthogonal vectors.

Understanding

Students understand that:

The dot product of two vectors is the sum of the products of their horizontal components and their vertical components.
If ? = ?1? + ?1? and ? = ?2? + ?2?, the dot product of ? and ? is defined by ? ? ? = ?1?2 + ?1?2.
Alternative Formula for the Dot Product: ? ? ? = ??? ??? cos ?, where ? is the smallest non negative angle between v and w.
Two vectors are orthogonal when the angle between them is 90^o. To show that two vectors are orthogonal, show that their dot product is zero.
A vector may be expressed as the sum of two orthogonal vectors, called the vector components.

Vocabulary

dot product
parallel
orthogonal
components
vector projection
vector components
decomposition

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