Calculus 3 - Vector Projections & Orthogonal Components

TL;DR

Learn how to find the parallel and perpendicular components of a vector with respect to another vector.

Transcript

let's say we have two vectors vector u and vector v how can we find two components of vector u one of which is parallel to vector v and the other is perpendicular to vector v so let's draw a picture so let's say this is w1 and this perpendicular to it is w2 how can we find those two components of vector u where w one is parallel to vector v and uh ... Read More

Key Insights

• 🫥 The parallel component of a vector to another vector can be found using the dot product and the magnitude of the parallel vector.
• ❓ The perpendicular component can be obtained by subtracting the parallel component from the original vector.
• ⛔ The components w1 and w2 are not limited to any specific axis and can be found using the provided formulas.
• ❎ The magnitude of vector v is calculated using the square root of the sum of the squares of its components.
• ❓ Practice problems are helpful in understanding and applying these concepts in different scenarios.
• 🤪 The process can be repeated for vectors with x, y, and z components.
• 😄 The magnitude of vector w2 will be the same as vector u, as it represents the component orthogonal or perpendicular to vector v.

Questions & Answers

Q: How do you find the parallel component of a vector to another vector?

To find the parallel component (w1), use the formula w1 = (u · v) / ||v||^2 * v, where · represents the dot product and ||v|| represents the magnitude of vector v.

Q: What is the formula to find the perpendicular component of a vector to another vector?

The formula to find the perpendicular component (w2) is w2 = u - w1, where w1 is the parallel component.

Q: Can the components of vector u be along any axis?

Yes, the components w1 and w2 are not necessarily along the x or y axis. They are simply parallel and perpendicular to vector v.

Q: How can the magnitude of vector v be calculated?

The magnitude of vector v, denoted as ||v||, is calculated as the square root of the sum of the squares of its x, y, and z components.

Q: What does w1 represent in relation to vector u and vector v?

w1 is the projection of vector u onto vector v. It represents the component of vector u that travels along vector v.

Q: Is vector v the same length as w1?

No, vector v is not necessarily the same length as w1. It is simply parallel to w1 and perpendicular to w2.

Q: How do you calculate w2?

w2 is calculated by subtracting w1 from vector u. The formula is w2 = u - w1.

Q: Can the formulas for w1 and w2 be used for any vectors?

Yes, the formulas for w1 and w2 can be used for any given vectors u and v.

Summary & Key Takeaways

• To find the parallel component of vector u to vector v (w1), use the formula w1 = (u · v) / ||v||^2 * v.

• To find the perpendicular component of vector u to vector v (w2), use the formula w2 = u - w1.

• Practice problems involve finding w1 and w2 using given vectors.