If a&b are vectors such that a+3b=-3i+j & a-b=i-2j then find vectors a&b.
subtract the equations:
4b=-4i+3j
solve for b. Then, solve the for a, using this solution for b.
Just algebra involved.
To find the vectors a and b, we will use a system of equations. Here's how we can solve it step by step:
Step 1: Write the given equations in the standard form.
a + 3b = -3i + j ---(Equation 1)
a - b = i - 2j ---(Equation 2)
Step 2: Solve Equation 2 for a in terms of b.
a = b + i - 2j ---(Equation 3)
Step 3: Substitute Equation 3 into Equation 1.
(b + i - 2j) + 3b = -3i + j
Step 4: Simplify the equation.
4b - 2i - 5j = -3i + j
Step 5: Group like terms.
4b - 2i + 5j = j - 3i
Step 6: Equate the coefficients of the i and j terms on both sides.
4b - 2i = -3i ----(Equation A)
5j = j ----(Equation B)
Step 7: Solve Equation B for j.
5j = j
4j = 0
j = 0
Step 8: Substitute the value of j back into Equation B.
5(0) = 0
0 = 0 (True)
Step 9: Since j = 0, we can substitute it into Equation A.
4b - 2i = -3i
Step 10: Move the variables to one side.
4b = -3i + 2i
Step 11: Simplify.
4b = -i
Step 12: Multiply both sides by (1/4) to solve for b.
b = (-1/4)i
Step 13: Substitute the value of b into Equation 3.
a = (-1/4)i + i - 2(0)
a = (3/4)i
So, the vectors a and b are:
a = (3/4)i
b = (-1/4)i