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ZingPath: Inner Product and Linear Dependence

Euclidean Inner Product

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Inner Product and Linear Dependence

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Euclidean Inner Product

Algebra-2

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You will learn and apply the Euclidean inner product of vectors and its properties.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Know the Euclidean inner product of vectors.
  • Calculate the Euclidean inner product of vectors.
  • Know that the Euclidean inner product of vectors is symmetric.
  • Know that the Euclidean inner product of vectors is bilinear.
  • Know that the Euclidean inner product of vectors is positive definite.

Everything You'll Have Covered

Recall that vectors are quantities consisting of direction and magnitude. They are represented in the plane by directed line segments, and written algebraically using Cartesian coordinates.

The Euclidean inner product is an operation on two vectors that returns a scalar. It is a very deep concept with many applications in physics. For example, if a force is applied upon an object in the same direction the object moves, then the work required to move the object is the product of the force and the distance the object moves. However, if the force and displacement have different directions, work is the Euclidean inner product of the two. In addition, the Euclidean inner product can be useful for solving geometric problems. If we know the length of the kite and the angle the kite is making with the ground in the picture below, we can find the length of its shadow using the Euclidean inner product.

Some important properties of the Euclidean inner product follow. Proofs for these properties are given in the Activity Object using both the algebraic formula and the geometric definition for Euclidean inner product.

Tutorial Details

Approximate Time 20 Minutes
Pre-requisite Concepts Students should know the definitions of vector and the direction of a vector, and be able to write the components of a vector on the coordinate plane.
Course Algebra-2
Type of Tutorial Concept Development
Key Vocabulary bilinear property, Euclidean inner product of vectors, positive definite property