Question

Let u = 2i + j, v = i + j, and w = i - j. Find scalars a and b such that u= av + bw.

Answer 1

Answer:

1/2 and 3/2

Step-by-step explanation:

Given the expression u= av + bw where u = 2i + j, v = i + j, and w = i - j, to find the scalars a and b, we will first substitute the vectors into the expression given as shown;

u= av + bw

2i + j = a( i + j)+b(i-j)

open the parenthesis

2i + j = ai+aj+bi-bj

collect like terms at the right hand side of the equation

2i + j = ai+bi+aj-bj

2i + j = (a+b)i+(a-b)j

compare the coefficient of complex value i and j on both sides

a+b = 2 ............... 1

a-b = 1................. 2

solve the resulting equation simultaneously using elimination method

Subtract both equation from each other.

(a-a) +(b-(-b))= 2-1

0+2b = 1

b = 1/2

Substitue b = 1/2 into equation 2 to get a

From equation 2, a = 1+b

a = 1 + 1/2

b = 3/2

Hence the scalars a and b are 1/2 and 3/2 respectively

Answer 2

$a=\frac{3}{2},b=\frac{1}{2}$

A complex number is of the form $\boldsymbol{a+ib}$ where $a,b$ are real numbers.

$u=2i+j$

$v=i+j$

$w=i-j$

$u=av+bw$

So,

$2i+j=a(i+j)+b(i-j)$

$2i+j=(a+b)i+(a-b)j$

Therefore,

$2=a+b\\1=a-b$

Add these equations.

$3=2a\\\frac{3}{2}=a$

So,

$1=\frac{3}{2}-b\\\\b=\frac{3}{2}-1$

  $=\frac{1}{2}$

For more information:

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