Vectors - Projection Of Vectors

Vectors - Projection of Vectors

Introduction to vectors and their properties
Understanding the concept of projection
Definition of projection of a vector onto another vector
Geometric interpretation of projection
Expression of the projection of vector ‘a’ onto vector ‘b’ as
- Proj(a,b) = (a · b) / ||b||^2 * b
Example 1: Suppose vector a = 2i + 3j + k and vector b = i - j + 2k. Find the projection of vector a onto vector b.
Example 2: Determine the projection of vector a = 3i - 2j onto vector b = 4i + 5j.
Orthogonal projection and its properties
Calculation of orthogonal projection using dot product "

Vector projection in physics and engineering applications
Finding the component of a force in a specified direction
Calculating work done by a force along a given direction
Determining the magnitude and direction of a force needed to produce a desired effect
Example: A 15 kg crate is being pulled with a force of 100 N at an angle of 30 degrees with the horizontal. Find the horizontal and vertical components of the force.

Projection of a vector onto the x-axis and y-axis
Expressing a vector in terms of its components
Finding the magnitude and direction of the projection
Example: Consider vector u = 3i + 4j. Find the projections of u onto the x-axis and y-axis.

Introduction to projection of a vector onto a plane
Determining the projection of a vector onto a plane
Definition of the normal vector of a plane
Calculating the projection using dot product and normal vector
Example: Find the projection of vector v = 2i - j + 3k onto the plane with equation 2x + 3y - z = 4.

Understanding orthogonal projection onto a plane
Definition and properties of orthogonal projection
Calculation of orthogonal projection using dot product and normal vector
Example: Determine the orthogonal projection of vector a = 4i + 2j - k onto the plane with equation 3x + 4y - 2z = 5.

Extending the concept of vector projection to three dimensions
Finding the projection of a vector onto a line in three dimensions
Determining the projection of a vector onto a plane in three dimensions
Example: Consider vector v = i - 2j - 3k. Find the projection of v onto the line with direction vector 2i + j - k.

Introduction to scalar projection
Definition of scalar projection
Calculation of scalar projection using dot product
Example: Find the scalar projection of vector a = 3i + 4j onto the vector b = 2i - j.

Relationship between vector projection and linear transformation
Understanding the transformation matrix for vector projection
Applying vector projection as a linear transformation
Example: Consider vectors u = 3i + 2j and v = 4i - j. Find the transformation matrix for the projection of u onto v.

Properties of vector projection
Commutative property: Proj(a, b) = Proj(b, a)
Distributive property: Proj(a, b + c) = Proj(a, b) + Proj(a, c)
Example: Prove the commutative and distributive properties of vector projection.

Relationship between vector projection and vector addition
Expressing the sum of two vectors in terms of their projections
Example: Let a, b, and c be vectors. Show that a - Proj(a, b) = Proj(a, c), where c = b + (a - Proj(a, b)).

Recap of vector projection onto a plane in three dimensions
Example: Find the projection of vector v = i + j + k onto the plane with equation x - 2y + z = 5.

Recap of scalar projection and its calculation using dot product
Example: Determine the scalar projection of vector a = 4i - 2j + 3k onto vector b = 3i + j - k.

Vector projection as a linear transformation
Defining the transformation matrix for vector projection
Example: Let a = 2i + j and b = 3i - k. Find the transformation matrix for the projection of a onto b.

Properties of vector projection: Magnitude relation
Relationship between the magnitude of the projection and the original vector
Magnitude of the projection as a fraction of the original vector
Example: Show that the magnitude of the projection of vector a onto vector b is less than or equal to the magnitude of vector a.

Properties of vector projection: Orthogonality
Understanding the orthogonality of the projection and the difference vector
Difference vector as the component orthogonal to the projection
Example: Prove that the difference vector between a vector and its projection is orthogonal to the projection vector.

Applications of vector projection in physics: Kinematics
Calculation of displacement, velocity, and acceleration components
Example: A particle moves with a velocity vector v = 4i - 2j - 3k. Determine its components parallel and perpendicular to the vector a = 2i + j.

Applications of vector projection in physics: Work and Energy
Determining the work done by a force in a specified direction
Calculation of the component of a force along a given direction
Example: A force of 50 N makes an angle of 60 degrees with the vertical. Find the work done in lifting an object vertically through 10 m.