For what value(s) of h is b in the plane spanned by a1 and a2?

Question

a1 = [1, 4, -2] a2 = [-2, -3, 7] b = [4, 1, h]

Answer 1

Explanation :

For the plane spanned by a1 and a2, y is a linear combination of a1 and a2 if it is in the plane.

y = a a1 + b a2 where a and b are constants.

[4,1,h] = a[1,4,-2] + b[-2,-3,7]

[4,1,h] = [a - 2b,4a -3b,-2a+ 7b] by the rules of vector addition and multiplication. Vectors are equal if their components are equal:

4 = a - 2b

1 = 4a - 3b

h = -2a + 7b

We can solve for a and b from the first two equations to find h:

a = -2

b = -3

h = -17

We see that -2[1,4,-2] - 3[-2,-3,7] = [4,1,-17].

Answer :

When h=-17 is b in the plane spanned by a1 and a2.

If you like the way I have answered this question, then please leave a comment and send me a message letting me know you liked it ! You can take a look at over other problems I have solved here on Wyzant Ask an Expert using the link on my tutor page. Please contact me about setting up an online session to help you achieve your best potential test scores and grades in Mathematics, Statistics, and Physics.