Understanding Linear Combinations and Planes

The video tutorial explains how to determine the value of h for which vector w is in the plane spanned by vectors u and v. It involves expressing vector w as a linear combination of vectors u and v, setting up a system of equations, and solving it using matrix operations. The process includes forming an augmented matrix, performing row operations to achieve reduced row echelon form, and substituting the found values to determine h. The solution shows that h must be -17 for vector w to be a linear combination of vectors u and v.

What is the main problem discussed in the video?

• Determining if vectors u and v are orthogonal.

• Calculating the magnitude of vector w.

• Finding the value of h for which vector w is in the plane spanned by vectors u and v.

• Finding the angle between vectors u and v.

What does it mean for vector w to be a linear combination of vectors u and v?

• Vector w is parallel to vectors u and v.

• Vector w is the cross product of vectors u and v.

• Vector w is perpendicular to vectors u and v.

• Vector w can be expressed as a sum of scalar multiples of vectors u and v.

How many equations are formed to express vector w as a linear combination of vectors u and v?

• Four equations

• Three equations

• Two equations

• One equation

What is the purpose of solving the first two equations in the system?

• To find the value of h directly.

• To determine if vectors u and v are linearly independent.

• To find the scalars c1 and c2.

• To check if vector w is orthogonal to vectors u and v.

What operation is performed to simplify the augmented matrix?

• Matrix subtraction

• Matrix multiplication

• Matrix addition

• Row reduction

What is the value of c1 found in the solution?

• 2

• 3

• -3

• -2

What is the value of c2 found in the solution?

• 3

• -3

• -2

• 2

How is the value of h determined?

• By solving the first equation.

• By calculating the dot product of vectors u and v.

• By finding the cross product of vectors u and v.

• By substituting c1 and c2 into the third equation.

What is the final value of h that makes vector w a linear combination of vectors u and v?

• -17

• 17

• 1

• 0

What conclusion is reached about vector w?

• Vector w is a linear combination of vectors u and v when h is -17.

• Vector w is parallel to vectors u and v.

• Vector w is not in the plane spanned by vectors u and v.

• Vector w is perpendicular to vectors u and v.