The least positive integral value of α, for which the angle between the vectors αi - 2j + 2k and αi

Q: Wait, why do we use the dot product? Can't we just use the cross product?

Great question! The cross product is primarily used to find a vector perpendicular to two others or to calculate the area of a parallelogram. For angles, the dot product is your best friend because it directly relates to the cosine of the angle, which is exactly what we need to check if the angle is acute ($\cos \theta > 0$).

Q: Why is the condition $\vec{a} \cdot \vec{b} > 0$ and not $\ge 0$?

The problem specifies an 'acute' angle. An angle of $90^\circ$ (where the dot product is exactly $0$) is a right angle, not an acute one. Therefore, we must have a strictly positive dot product to ensure the angle is less than $90^\circ$.

Q: Why do we reject the negative interval when looking for the least positive integer?

The question explicitly asks for the 'least positive integral value' of $\alpha$. While our inequality $\alpha^2 - 4\alpha - 4 > 0$ holds for $\alpha 2 + 2\sqrt{2}$.

Question

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha\hat{i} - 2\hat{j} + 2\hat{k}$ and $\alpha\hat{i} + 2\alpha\hat{j} - 2\hat{k}$ is acute, is_____

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in a three-dimensional space, holding two vectors, $\vec{a} = \alpha\hat{i} - 2\hat{j} + 2\hat{k}$ and $\vec{b} = \alpha\hat{i} + 2\alpha\hat{j} - 2\hat{k}$. These are dynamic entities that shift as we vary the parameter $\alpha$.

Our mission is to find the smallest positive integer $\alpha$ that forces these two vectors to form an acute angle. This is a fundamental problem of geometry and algebra.

### The Golden Rule of Dot Products

To determine the angle between two vectors, we utilize the dot product:
$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos 	heta$$

Here, $	heta$ represents the angle between the vectors. For the angle to be acute, we require $\cos 	heta > 0$.

Since the magnitudes $|\vec{a}|$ and $|\vec{b}|$ are always non-negative, the sign of $\cos 	heta$ is determined solely by the sign of the dot product. Thus, our condition for an acute angle is simply $\vec{a} \cdot \vec{b} > 0$.

### The Algebraic Battle

We calculate the dot product by multiplying the corresponding components:
$$(\alpha)(\alpha) + (-2)(2\alpha) + (2)(-2) > 0$$

Simplifying this expression, we obtain the quadratic inequality:
$$\alpha^2 - 4\alpha - 4 > 0$$

To solve this, we find the roots of the equation $\alpha^2 - 4\alpha - 4 = 0$ using the quadratic formula:
$$\alpha = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Substituting our values, we get:
$$\alpha = \frac{4 \pm \sqrt{16 - 4(1)(-4)}}{2} = \frac{4 \pm \sqrt{32}}{2}$$

This simplifies to:
$$\alpha = 2 \pm 2\sqrt{2}$$

### The Final Decision

With our roots at $2 - 2\sqrt{2}$ and $2 + 2\sqrt{2}$, we analyze the number line. Given that $\sqrt{2} \approx 1.414$, our roots are approximately $-0.828$ and $4.828$.

Because the parabola $\alpha^2 - 4\alpha - 4$ opens upwards, the expression is positive outside the roots. We require $\alpha$ to be in the interval $(-\infty, -0.828) \cup (4.828, \infty)$.

The problem specifically demands the least positive integral value. Since the negative interval is irrelevant, we look at the positive interval where $\alpha > 4.828$.

The smallest integer satisfying this condition is clearly **$5$**. We have successfully navigated the geometry and conquered the algebra to find that the least positive integral value of $\alpha$ is **$5$**.

Animated Solution for Mathematics - Vector Algebra: The least positive integral value of $α$ , for which the angle between the vectors $α \hat{i} - 2 \hat{j} + 2 \hat{k}$ and $α \hat{i} + 2 α \hat{j} - 2 \hat{k}$ is acute, is_____

Enter Numerical Value:

Visualized Solution

Given Vectors

Condition for Acute Angle

Dot Product Setup

Simplifying the Inequality

Finding the Roots

Calculating the Roots

Numerical Approximation

Analyzing the Inequality

Finding the Least Positive Integer

The Sigma Insight: Scalar (Dot) Product

Analyzing the Setup

The Golden Rule of Dot Products

The Algebraic Battle

The Final Decision

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