Question

The values of $\alpha$ so that $|\alpha\hat{i}+(\alpha+1)\hat{j}+2\hat{k}|=3$, are

• A. 2, -4
• B. 1, 2
• C. -1, 2
• D. -2, 4
• E. 1, -2

Hint:

Algebra Tip: Always check if you can divide the entire quadratic equation by a common constant (like dividing by 2 here). It makes factoring significantly faster and reduces arithmetic mistakes.

Answer 1

Answer: (E)

Concept:
The magnitude of a vector $\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}$ is given by $|\vec{v}| = \sqrt{a^2 + b^2 + c^2}$. We can equate this to the given magnitude and solve the resulting polynomial equation to find the unknown variable.

Step 1: Set up the magnitude equation.
Given the vector and its magnitude $|\alpha\hat{i}+(\alpha+1)\hat{j}+2\hat{k}| = 3$, apply the magnitude formula: $$\sqrt{\alpha^2 + (\alpha+1)^2 + 2^2} = 3$$

Step 2: Square both sides to remove the radical.
To eliminate the square root, square the entire equation: $$\alpha^2 + (\alpha+1)^2 + 4 = 9$$

Step 3: Expand the binomial term.
Expand $(\alpha+1)^2$ using $(a+b)^2 = a^2 + 2ab + b^2$: $$\alpha^2 + (\alpha^2 + 2\alpha + 1) + 4 = 9$$

Step 4: Form a standard quadratic equation.
Combine like terms and move everything to the left side to set it to zero: $$2\alpha^2 + 2\alpha + 5 = 9$$ $$2\alpha^2 + 2\alpha - 4 = 0$$

Step 5: Factor and solve for $\alpha$.
Divide the entire equation by 2 to simplify: $$\alpha^2 + \alpha - 2 = 0$$ Factor the quadratic equation: $$(\alpha + 2)(\alpha - 1) = 0$$ Setting each factor to zero yields the solutions $\alpha = -2$ and $\alpha = 1$. Hence the correct answer is (E) 1, -2.