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

Answer:

(a) w = $(\frac{-7}{4} , \frac{9}{4}, \frac{1}{2})$

(b) -2u + 3v - 5w = ($\frac{39}{4}$, $\frac{-54}{4}$, $\frac{3}{2}$)

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 = $\frac{-7}{4}$

From (**)

4b = 11 - 2

4b = 9

b = $\frac{9}{4}$

From (***)

-1 - 4c = -3

4c = -1 + 3

4c = 2

c = $\frac{2}{4}$

c = $\frac{1}{2}$

Remember that

w = (a, b, c)

w = ai + bj + ck

Therefore,

w = $(\frac{-7}{4} , \frac{9}{4}, \frac{1}{2})$

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

Remember that;

u = (−2, 3, 1)

v = (−1, −1, 2),

w = $(\frac{-7}{4} , \frac{9}{4}, \frac{1}{2})$

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

-2(−2, 3, 1) + 3(−1, −1, 2) - 5$(\frac{-7}{4} , \frac{9}{4}, \frac{1}{2})$

Expand

(4, -6, -2) + (−3, −3, 6) - $(\frac{-35}{4} , \frac{45}{4}, \frac{5}{2})$

Collect like terms

(4-3+$\frac{35}{4}$, -6-3-$\frac{45}{4}$, -2+6-$\frac{5}{2}$)

Solve

($\frac{39}{4}$, $\frac{-54}{4}$, $\frac{3}{2}$)

Therefore, -2u + 3v - 5w = ($\frac{39}{4}$, $\frac{-54}{4}$, $\frac{3}{2}$)