Q.The value of 𝛼 if the angle between 𝑝⃗ = 2𝛼2𝑖̂ − 3𝛼𝑗̂ + 𝑘̂ and 𝑞⃗ = 𝑖̂ + 𝑗̂ + 𝛼𝑘̂ is obtuse, is
(A) 𝑅 − [0, 1]
(B) (0, 1)
(C) [0, ∞)
(D) [1, ∞)

Concept: Dot Product Sign — For an obtuse angle, $\vec{p} \cdot \vec{q} < 0$.

Step 1: Compute the dot product:

Step 2: Set the inequality for an obtuse angle:

$2\alpha^2 - 2\alpha < 0 \quad\Rightarrow\quad 2\alpha(\alpha - 1) < 0.$ …

The key to this problem lies in a single, elegant fact: the sign of the dot product tells you everything about the angle between two vectors. If the dot product is positive, the angle is acute (less than 90°). If it's zero, the vectors are perpendicular. If it's negative, the angle is obtuse — greater than 90° and less than 180°.

So when a problem says "the angle is obtuse," it's handing you a direct inequality: $\vec{p} \cdot \vec{q} < 0$. No need to compute the angle itself, no inverse cosine, no magnitudes — just the dot product and a simple inequality.

Let's apply this.

1. Write down the vectors clearly.

   $\vec{p} = 2\alpha^2 \hat{i} - 3\alpha \hat{j} + 1 \hat{k}$

   $\vec{q} = 1 \hat{i} + 1 \hat{j} + \alpha \hat{k}$

2. Compute the dot product.

   Multiply corresponding components and add:

   $\vec{p} \cdot \vec{q} = (2\alpha^2)(1) + (-3\alpha)(1) + (1)(\alpha)$

   $= 2\alpha^2 - 3\alpha + \alpha$

   $= 2\alpha^2 - 2\alpha$

3. Set up the obtuse condition.

   For an obtuse angle, $\vec{p} \cdot \vec{q} < 0$:

   $2\alpha^2 - 2\alpha < 0$

4. Solve the quadratic inequality.

   Factor out the common term:

   $2\alpha(\alpha - 1) < 0$

   Divide both sides by 2 (positive, so inequality direction stays the same):

   $\alpha(\alpha - 1) < 0$

   This is a simple quadratic inequality. The product of two numbers is negative when one is positive and the other is negative. The roots of $\alpha(\alpha - 1) = 0$ are $\alpha = 0$ and $\alpha = 1$.

   For a quadratic with a positive leading coefficient, the expression is negative between the roots.

   So $\alpha(\alpha - 1) < 0$ when $0 < \alpha < 1$. …

Method: Using the sign of the dot product to pin down a parameter

Use this whenever a problem states that the angle between two vectors is acute, right, or obtuse and asks for the values of an unknown appearing in them.

Steps

Step 1: Translate the angle condition into a sign condition.

Since $\vec{p}\cdot\vec{q}=|\vec{p}||\vec{q}|\cos\theta$ and the magnitudes are positive, the sign of the dot product equals the sign of $\cos\theta$:

Step 2: Form the dot product as a function of the parameter. …

Common Mistakes

Mistake 1: Using $\le 0$ instead of $<0$ for "obtuse".

Why it's wrong: a zero dot product means a $90^\circ$ (right) angle, which is not obtuse, so the endpoints must be excluded. Correct approach: solve $\vec{p}\cdot\vec{q}<0$ strictly, giving the OPEN interval $(0,1)$.

Mistake 2: Mishandling the $\hat{k}$ terms in the dot product. …