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?
Answer:
x = ___ cm
y = ___ cm
Answer:
x = 7.0cm
y = -8.27cm
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.