Question

Suppose that →p,→q and →r are three non-coplanar vectors in R³ . Let the components of vector of a vector →s along →p,→q and →r be 4, 3 and 5 respectively. If the components of this vector →s along (−→p + →q + →r) , (→p − →q + →r) and (−→p − → q + → r) are x,y and z respectively , then the value of 2x+y+z is

Answer 1

The value of 2x + y + z is 12, where x, y, and z are the components of the vector →s along the vectors (-→p + →q + →r), (→p - →q + →r), and (-→p - →q + →r).

Step-by-step explanation:

Given that →s can be represented as the sum of its components along →p, →q, and →r, which are 4, 3, and 5, respectively, we can write →s = 4→p + 3→q + 5→r. The task is to find the components of →s along the vectors (-→p + →q + →r), (→p - →q + →r), and (-→p - →q + →r), denoted by x, y, and z respectively.

Each of these new vectors can be expressed in terms of the components along →p, →q, and →r:

• x = -4 + 3 + 5 = 4
• y = 4 - 3 + 5 = 6
• z = -4 - 3 + 5 = -2

Now we can find the value of 2x + y + z which is:

2x + y + z = 2*4 + 6 + (-2) = 8 + 6 - 2 = 12.

Thus, the value of 2x + y + z is 12.