Vector has x and y components of -8.80 cm and 18.0 cm, respectively; vector has x and y components of 12.2 cm and -6.80 cm, respectively. If - + 3 = 0, what are the components of ? x = cm y = cm

Question:

Vector A has x and y components of −8.80 cm  and 18.0 cm , respectively; vector B has x and  y components of 12.2 cm and −6.80 cm , respectively.  If A − B +3 C = 0, what are the components of C?

x = ___ cm

y = ___ cm

x = 7.0cm

y = -8.27cm

Step-by-step explanation:

For a vector F, with x and y components of a and b respectively, its unit vector representation is as follows;

F = ai + bj              [Where i and j are unit vectors in the x and y directions respectively]

Using this analogy, let's represent vectors A and B from the question in their unit vector notation.

A has an x-component of -8.80cm and y-component of 18.0cm

B has an x-component of 12.2cm and y-component of -6.80cm,

In unit vector notation, these become;

A = -8.80i + 18.0j

B = 12.2 i + (-6.80)j = 12.2i - 6.80j

Also, there is a third vector C. Let the x and y components of C be a and b respectively. Therefore,

C = ai + bj

Now,

A - B + 3C = 0                [substitute the vectors]

=> [-8.80i + 18.0j] - [12.2 i -6.80j] + [3(ai + bj)] =  0        [open brackets]

=> -8.80i + 18.0j - 12.2 i + 6.80j + 3(ai + bj) =  0

=> -8.80i + 18.0j - 12.2 i + 6.80j + 3ai + 3bj =  0

=> -8.80i + 18.0j - 12.2 i + 6.80j + 3ai + 3bj =  0  [collect like terms and solve]

=> -8.80i  - 12.2 i  + 3ai + 6.80j + 18.0j + 3bj =  0

=> -21.0 i  + 3ai + 24.8j + 3bj =  0       [re-arrange]

=> 3ai + 3bj = 21.0i - 24.8j

Comparing both sides shows that;

3a = 21.0  (i)

3b = -24.8    (ii)

From equation (i)

3a = 21.0

a = 21.0 / 3 = 7.0

From equation (ii)

3b = -24.8

b = -24.8 / 3

b = -8.27

Therefore, the x-component and y-component of vector B which are a and b, are 7.0cm and -8.27cm respectively.