Components of a vector | MIT 18.02SC Multivariable Calculus, Fall 2010

TL;DR

Compute the component of one vector in the direction of another by using the dot product formula and the length of the direction vector.

Transcript

JOEL LEWIS: Hi. Welcome back to recitation. In lecture, among other things, you've been learning about computing components of one vector in the direction of another vector. So I have a straightforward problem about that for you here. So we've got two vectors. The vector 2i minus 2j plus k. And we've got the vector i plus j plus k. And so what I'd ... Read More

Key Insights

• 💻 Computing the component of one vector in the direction of another involves finding the length of the vector's projection onto the direction vector.
• ❎ The component can be positive or negative, depending on whether the projection is in the same or opposite direction as the direction vector.
• 🫥 The component formula can be derived from the dot product formula and the length of the direction vector.
• 🫥 The component formula simplifies the calculation process by using the dot product and the length of the direction vector.
• 💁 The component formula is applicable when vectors are given in their coordinate forms.
• 🫥 The dot product of two vectors can be calculated by multiplying their corresponding components and summing them.
• ❎ The length of a vector can be determined by using the square root of the sum of the squares of its components.

Questions & Answers

Q: What is the definition of the component of one vector in the direction of another vector?

The component of one vector in the direction of another is the length of the vector's projection onto the direction vector, with a sign if necessary.

Q: How is the component calculated?

The component can be calculated by taking the dot product of the two vectors and dividing it by the length of the direction vector.

Q: How is the length of the direction vector determined?

The length of the direction vector can be calculated using the usual length formula for vectors.

Q: How can the component formula be used in practice?

To find the component of a specific vector in the direction of another vector, calculate the dot product of the two vectors and the length of the direction vector, and then use them in the component formula.

Summary & Key Takeaways

• The component of one vector in the direction of another is the length of the projection of the vector onto the direction vector, with a sign if necessary.

• The component can be calculated using the dot product formula and the length of the direction vector.

• To find the component of a specific vector in the direction of another vector, the dot product of the two vectors and the length of the direction vector need to be calculated and used in the formula.