Question

If \(\bar{a}=2\hat{i}-\hat{j}+\hat{k}\), \(\bar{b}=\hat{i}+3\hat{j}-\hat{k}\), \(\bar{c}=-2\hat{i}+\hat{j}-3\hat{k}\) and \(\bar{d}=3\hat{i}+2\hat{j}+5\hat{k}\), then the value of \(x^{2}+z^{2}-y\) is \(\cdots\), where \(\bar{d}=x\bar{a}+y\bar{b}+z\bar{c}\)

• A. \(\frac{35}{4}\)
• B. \(\frac{17}{2}\)
• C. \(\frac{33}{4}\)
• D. \(\frac{15}{2}\)

Hint:

Equate the components of d with those of x a plus y b plus z c to get three linear equations.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Write \(\bar d=x\bar a+y\bar b+z\bar c\) and compare coefficients of \(\hat i,\hat j,\hat k\).

Step 2: Form the Equations
\(\hat i\): \(2x+y-2z=3\)
\(\hat j\): \(-x+3y+z=2\)
\(\hat k\): \(x-y-3z=5\)

Step 3: Solve
Adding the last two: \(2y-2z=7\), so \(y=z+\dfrac72\).\nFrom the third: \(x=5+y+3z=\dfrac{17}{2}+4z\).\nSubstituting in the first: \(\dfrac{41}{2}+7z=3\) gives \(z=-\dfrac52\), \(y=1\), \(x=-\dfrac32\).

Step 4: Calculation
\(x^{2}+z^{2}-y=\dfrac94+\dfrac{25}{4}-1=\dfrac{15}{2}\).

Final Answer:
The value of x squared plus z squared minus y is 15 over 2. \[ \boxed{\text{(D) }\dfrac{15}{2}} \]