Question

The coordinates of (4,5,6) with respect to the basis set $e_{1}=(1,1,1), e_{2}=(-1,1,0), e_{3}=(1,0,-1)$ are}

• A. (1,1,6)
• B. (2,1,3)
• C. (5,0,-1)
• D. (3,2,3)

Hint:

Coordinate vectors represent how much of each basis "ingredient" is needed to build the target vector.

Answer 1

The Correct Option is C

Step 1: Concept
To find the coordinates $(c_1, c_2, c_3)$, we solve the vector equation $c_1e_1 + c_2e_2 + c_3e_3 = (4,5,6)$.

Step 2: Meaning
This gives the system of equations: 1) $c_1 - c_2 + c_3 = 4$ 2) $c_1 + c_2 = 5$ 3) $c_1 - c_3 = 6$

Step 3: Analysis
From (3), $c_3 = c_1 - 6$. From (2), $c_2 = 5 - c_1$. Substitute into (1): $c_1 - (5 - c_1) + (c_1 - 6) = 4$. $c_1 - 5 + c_1 + c_1 - 6 = 4 \Rightarrow 3c_1 - 11 = 4 \Rightarrow 3c_1 = 15 \Rightarrow c_1 = 5$.

Step 4: Conclusion
Now find others: $c_2 = 5 - 5 = 0$; $c_3 = 5 - 6 = -1$. The coordinates are (5,0,-1). Final Answer: (C)