Question

Let \(\bar{a}=\lambda \hat{i}+4\hat{j}\) and \(\bar{b}=\hat{i}+\lambda \hat{j}\) where \(\lambda \in Z\) such that \(|\bar{a}|\lt 6\). The number of possible values of \(\lambda\) for which \(\bar{a}\) and \(\bar{b}\) are non-collinear is ___

• A. \(7\)
• B. \(9\)
• C. \(3\)
• D. \(5\)

Hint:

Use |a| < 6 to list the integers, then remove the values making the vectors parallel.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
\(|\bar{a}|<6\) means \(\lambda ^{2}+16<36\), so \(\lambda ^{2}<20\).

Step 2: List the integers
\(\lambda \in \{-4,-3,-2,-1,0,1,2,3,4\}\), which is 9 values.

Step 3: Collinear case
\(\bar{a}\parallel \bar{b}\) needs \(\dfrac{\lambda }{1}=\dfrac{4}{\lambda }\), so \(\lambda ^{2}=4\), giving \(\lambda =\pm 2\).

Step 4: Count
Non-collinear values: \(9-2=7\).

Final Answer:
There are \(7\) possible values of \(\lambda\). \[ \boxed{\text{(A) }7} \]