Question

Let \(\bar{a}=2\hat{i}-\hat{j}+\hat{k}\) and \(\bar{b}=\hat{i}+2\hat{j}-2\hat{k}\). If a third vector \(\bar{c}=x\hat{i}+y\hat{j}+z\hat{k}\) is coplanar with \(\bar{a}\) and \(\bar{b}\), is perpendicular to \(\bar{a}\), and satisfies \(|\bar{c}|=\sqrt{45}\), then the sum of all possible values of \(y\) is:

• A. \(0\)
• B. \(2\)
• C. \(5\)
• D. \(-5\)

Hint:

A vector coplanar with a and b can be written as a combination of them. Impose perpendicularity to a and the given magnitude.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Since \(\bar c\) is coplanar with \(\bar a\) and \(\bar b\), write \(\bar c=\lambda\bar a+\mu\bar b\).

Step 2: Perpendicular to a
\(\bar c\cdot\bar a=0\) gives \(\lambda|\bar a|^{2}+\mu(\bar a\cdot\bar b)=0\).
Here \(|\bar a|^{2}=6\) and \(\bar a\cdot\bar b=2-2-2=-2\), so \(6\lambda-2\mu=0\) and \(\mu=3\lambda\).

Step 3: Find the Direction
\(\bar c=\lambda(\bar a+3\bar b)=\lambda(5\hat i+5\hat j-5\hat k)\).

Step 4: Use the Magnitude
\(|\bar c|^{2}=75\lambda^{2}=45\), so \(\lambda=\pm\sqrt{\dfrac35}\).

Step 5: Sum of y Values
\(y=5\lambda\), so the two values are \(\pm5\sqrt{\dfrac35}\), and their sum is \(0\).

Final Answer:
The sum of all possible values of y is 0. \[ \boxed{\text{(A) }0} \]