Question

if the angle between \(\underline{\mathrm{a}}=(2,-\mathrm{m}, 3 \mathrm{~m})\) and \(\underline{\mathrm{b}}=(1+\mathrm{m},-2 \mathrm{~m}, 1)\) is acute then \(\mathrm{m}=\) (a) \(\mathrm{m} \in \mathrm{R}\) (b) \(\mathrm{m} \in \overline{(-\infty,}-2) \cup[-(1 / 2), \infty]\) (c) \(\mathrm{m}=-(1 / 2)\) (d) \(m \in[-2,-(1 / 2)]\)

Short answer

\(m \in \left(-\infty, -\frac{1}{2}\right) \cup (2, \infty)\)

Step-by-step solution

Define the vectors and their dot product

The problem gives us the vectors \(\vec{a}=(2, -m, 3m)\) and \(\vec{b}=(1+m, -2m, 1)\). We begin by computing their dot product. The dot product of two vectors is defined as: $$ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3, $$ where \(a_1, a_2,\) and \(a_3\) are the components of \(\vec{a}\) and \(b_1, b_2,\) and \(b_3\) are the components of \(\vec{b}\).

Compute the dot product

Using the components of the given vectors, we find their dot product: $$ \vec{a} \cdot \vec{b} = (2)(1+m) + (-m)(-2m) + (3m)(1) = 2 + 2m - 2m^2 + 3m. $$

Apply the condition for an acute angle

The angle between two vectors is acute if their dot product is positive. In other words: $$ \vec{a} \cdot \vec{b} > 0. $$ Now, let's substitute the computed dot product for \(\vec{a} \cdot \vec{b}\): $$ 2 + 2m - 2m^2 + 3m > 0. $$

Simplify and solve the inequality

Next, we simplify the inequality: $$ 2 - 2m^2 + 5m > 0. $$ Now, we need to find the roots of the quadratic and check the interval in between: $$ m_1 = -(\frac{1}{2}) \quad \text{and} \quad m_2 = 2. $$ Hence the solution set for \(m\) is: $$ m \in \left(-\infty, -\frac{1}{2}\right) \cup (2, \infty). $$ So the correct answer is (b) \(m \in \left(-\infty, -\frac{1}{2}\right) \cup [-(1 / 2), \infty]\).

Key concepts

Dot Product

The dot product is a fundamental operation in vector algebra. It is the sum of the products of the corresponding components of two vectors. For vectors \( \vec{a} = (a_1, a_2, a_3) \) and \( \vec{b} = (b_1, b_2, b_3) \), the dot product is given by:\[ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3. \]

• This operation results in a scalar, not a vector.
• The dot product is a measure of how much one vector extends in the direction of another.

If the dot product is zero, the vectors are perpendicular. It is essential to calculate the dot product correctly, as it plays a key role in determining vector relationships, such as whether the angle between vectors is acute.

Acute Angle

An acute angle is an angle less than \( 90^\circ \). In the context of vectors, an acute angle between two vectors indicates that their dot product is positive. This is because:\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \]

• Here, \(|\vec{a}|\) and \(|\vec{b}|\) are the magnitudes of the vectors.
• \(\theta\) is the angle between them.

Since \( \cos(\theta) \) is positive for acute angles, the dot product will also be positive. Understanding this principle is crucial when determining vector relationships, such as when solving inequalities involving vector angles.

Quadratic Inequality

A quadratic inequality is an inequality involving a quadratic expression, such as \[ ax^2 + bx + c > 0. \]

• You solve it by finding the roots of the corresponding equation \( ax^2 + bx + c = 0 \).
• Use these roots to test intervals and determine where the inequality holds.

To solve a quadratic inequality like the one in the problem, you can follow these steps:1. Simplify the inequality.2. Find the roots using the quadratic formula: \[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]3. Test intervals defined by these roots to find where the inequality is satisfied.Knowing how to solve quadratic inequalities allows for evaluating vector conditions and relationships efficiently.

Vector Components

Vector components break a vector down into individual parts or dimensions. For example, a vector \( \vec{a} = (a_1, a_2, a_3) \) consists of components \( a_1 \), \( a_2 \), and \( a_3 \) along the x, y, and z axes respectively. Understanding these components is vital for:

• Performing operations, like addition, subtraction, and dot product.
• Calculating magnitudes and directions.

Each component affects the overall behavior of the vector in space. When working with vector problems, being able to manipulate and understand these components is critical to obtaining the correct results.