How to Find the Projection of u Onto v and the Vector Component of u Orthogonal to v (2 dimensions)

TL;DR

Explanation of finding projection of one vector onto another and the orthogonal component.

Transcript

in this video we're going to find the projection of u onto v and the vector component of u orthogonal to v just really quickly before we do the problem let me actually explain what this is it'll only take like 10 seconds so say we have a vector here and it's u we have another vector here and it's v so what are we finding in this problem so if you t... Read More

Key Insights

• 💦 Vector projection involves dropping one vector onto another to find the parallel component.
• 💁 The orthogonal component forms a 90-degree angle with the projected vector.
• 🤩 Dot product and magnitude play key roles in calculating vector projections.
• 🦻 Proper understanding of vector components aids in solving complex mathematical problems.
• ❓ Projections and orthogonal components are fundamental concepts in vector algebra.
• ❓ Formulas for projection and orthogonal components simplify vector calculations.
• ❓ Vector components can be calculated efficiently using formulaic approaches.

Questions & Answers

Q: What is the projection of one vector onto another, and how is it calculated?

The projection of one vector onto another, denoted as w1, is obtained by u.v / ||v||^2 * v, where u and v are the vectors involved.

Q: How is the vector component of one vector orthogonal to another determined?

The vector component of one vector orthogonal to another, denoted as w2, is found by subtracting the projection of u onto v from u.

Q: Why is understanding vector projections and orthogonal components important in mathematics?

Understanding these concepts helps in analyzing vector relationships, decomposing vectors, and solving various mathematical problems efficiently.

Q: How are the dot product and magnitude of vectors used in calculating projections and orthogonal components?

The dot product is used to determine the projection, while the magnitude of the vector is crucial for normalizing the projection formula.

Summary & Key Takeaways

• Explanation of finding the projection of vector u onto vector v by dropping it onto v, denoted as w1.

• Calculation of the vector component of u orthogonal to v, denoted as w2, forming a 90-degree angle.

• Formulas used: Projection of u onto v = u.v / ||v||^2 * v; Orthogonal component = u - projection.