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

A complex number is of the form where are real numbers.

So,

Therefore,

Add these equations.

So,

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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

d. millimeters per hour.

step-by-step explanation:

we are given that,

carlo used a rain gauge to measure the amount of water his lawn is receiving from the sprinklers.

since, according to the options, we see that,

water cannot be measured by 'miles per hour', 'meters per second' and 'grams per seconds'.

we get that, the water can be measured by 'millimeters per hour'.

hence, the appropriate rate for describing the amount of water is 'millimeters per hour'.