Question

Let u = (−2, 3, 1), v = (−1, −1, 2), and 3u − 2v − 4w = (3, 2, −3). Find: a. The vector w.b. -2u + 3v - 5w

Answer 1

To find vector w, rearrange the equation 3u - 2v - 4w = (3, 2, -3) and solve for w. To find -2u + 3v - 5w, substitute the values of u, v, and w into the expression and simplify.

Step-by-step explanation:

To find w, we can rearrange the equation 3u - 2v - 4w = (3, 2, -3) and solve for w. Step 1: Distribute the coefficients: 3u - 2v - 4w = 3u - 2v - 12w. Step 2: Combine like terms: -12w = (3, 2, -3) - 3u + 2v. Step 3: Divide both sides by -12: w = [(3, 2, -3) - 3u + 2v] / -12.

To find -2u + 3v - 5w, we substitute the values of u, v, and w into the expression and simplify. Step 1: Substitute the values: -2u + 3v - 5w = -2(-2, 3, 1) + 3(-1, -1, 2) - 5w. Step 2: Multiply and combine like terms: -2(-2, 3, 1) + 3(-1, -1, 2) - 5w = (4, -6, -2) + (-3, -3, 6) - 5w = (1, -9, 4) - 5w.

Answer 2

(a) w =
`((-7)/(4) , (9)/(4), (1)/(2))`

(b) -2u + 3v - 5w = (
`(39)/(4)`,
`(-54)/(4)`,
`(3)/(2)`)

Step-by-step explanation:

Given:

u = (−2, 3, 1)

=> u = -2i + 3j + k --------------------------(i)

v = (−1, −1, 2)

=> v = -i - j + 2k --------------------(ii)

3u − 2v − 4w = (3, 2, −3)

=> 3u − 2v − 4w = 3i + 2j - 3k ------------------(iii)

(A) TO FIND THE VECTOR w

Let:

w = (a, b, c) = ai + bj + ck

(a) Substitute u, v and w into equation (iii)

3u − 2v − 4w = 3i + 2j - 3k

3(-2i + 3j + k) - 2(-i - j + 2k) - 4(ai + bj + ck) = 3i + 2j - 3k

(b) Solve the equation in step (a) by opening the brackets and collecting like terms

(-6i + 9j + 3k) - (-2i - 2j + 4k) - (4ai + 4bj + 4ck) = 3i + 2j - 3k

open brackets

-6i + 9j + 3k + 2i + 2j - 4k - 4ai - 4bj - 4ck = 3i + 2j - 3k

collect like terms

-6i + 2i - 4ai + 9j + 2j - 4bj + 3k - 4k - 4ck = 3i + 2j - 3k

i(-4 - 4a) + j(11 - 4b) + k(-1 - 4c) = 3i + 2j - 3k

(c) Solve for a, b and c in step (b)

Comparing both sides of the equation, we have;

-4 - 4a = 3 ----------(*)

11 - 4b = 2 -----------(**)

-1 - 4c = -3 ------------(***)

From (*)

4a = -4 - 3

4a = -7

a =
`(-7)/(4)`

From (**)

4b = 11 - 2

4b = 9

b =
`(9)/(4)`

From (***)

-1 - 4c = -3

4c = -1 + 3

4c = 2

c =
`(2)/(4)`

c =
`(1)/(2)`

Remember that

w = (a, b, c)

w = ai + bj + ck

Therefore,

w =
`((-7)/(4) , (9)/(4), (1)/(2))`

(B) TO FIND -2u + 3v - 5w

Remember that;

u = (−2, 3, 1)

v = (−1, −1, 2),

w =
`((-7)/(4) , (9)/(4), (1)/(2))`

Substitute u, v, w into the expression as follows;

-2(−2, 3, 1) + 3(−1, −1, 2) - 5
`((-7)/(4) , (9)/(4), (1)/(2))`

Expand

(4, -6, -2) + (−3, −3, 6) -
`((-35)/(4) , (45)/(4), (5)/(2))`

Collect like terms

(4-3+
`(35)/(4)`, -6-3-
`(45)/(4)`, -2+6-
`(5)/(2)`)

Solve

(
`(39)/(4)`,
`(-54)/(4)`,
`(3)/(2)`)

Therefore, -2u + 3v - 5w = (
`(39)/(4)`,
`(-54)/(4)`,
`(3)/(2)`)