Question

If \(a^{2}+b^{2}+c^{2}=1\), then \(a b+b c+a c\) lies in the interval: (a) \([1,2 / 3]\) (b) \([-1 / 2,1]\) (c) \([-1,1 / 2]\) (d) \([2,-4]\)

Short answer

Answer: The expression \(ab + ac + bc\) lies in the interval \([-\frac{1}{2},1]\).

Step-by-step solution

Use the triangle inequality to bound the expression

We can bound the expression \(ab + ac + bc\) using the triangle inequality, which states that: \[\left|a+b\right|^2+\left|b+c\right|^2+\left|c+a\right|^2\geq0 \] Expanding the square terms, we have: \[(a^2 + 2ab + b^2) + (b^2 + 2bc + c^2) + (c^2 + 2ac + a^2) \geq 0\]

Re-arrange the inequality

Let's re-arrange the inequality by moving all terms involving \(ab\), \(ac\), \(bc\) to one side and the other terms to the other side: \[4(ab + ac + bc) + 2(a^2 + b^2 + c^2) \geq 0\] We are given that \(a^2 + b^2 + c^2 = 1\), so we can substitute this into our inequality: \[4(ab + ac + bc) + 2(1) \geq 0\]

Solve for the expression

We want to solve for the expression \(ab + ac + bc\), so let's re-arrange the inequality one more time: \[ab + ac + bc \geq -\frac{1}{2}\] This gives us a lower bound for the expression.

Use square terms for an upper bound

Now let's find an upper bound for the expression. Since \(a^2 + b^2 + c^2 = 1\), all the quantities \(a^2\), \(b^2\), \(c^2\) are non-negative. We know that the product of two non-negative numbers is non-negative; therefore: \[ab + ac + bc \leq a^2 + b^2 + c^2 = 1\] This gives us an upper bound for the expression.

Combine the bounds

We have found that the expression \(ab + ac + bc\) is bounded by: \[ -\frac{1}{2} \leq ab + ac + bc \leq 1\] Comparing this interval with the given options, we see that it matches option (b). Therefore, \(ab + ac + bc\) lies in the interval \([-\frac{1}{2},1]\).

Key concepts

Triangle Inequality

The triangle inequality theorem is a fundamental principle in geometry, often applied in various mathematical and real-world contexts. At its core, the theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. But its use isn't limited to geometrical shapes; it also lends a hand in inequalities involving algebraic expressions.

In the context of algebra, the triangle inequality helps us establish bounds for certain expressions. For instance, if we consider the modulus function, which gives the absolute value of a number, we can express the triangle inequality as \( |a + b| \leq |a| + |b| \). This can be decisive when we attempt to bound an expression composed of several terms, as in our example problem. It allows us to ensure the sum (or difference) of two numbers doesn't exceed specific limits.

By expanding this concept to algebraic variables and combining square terms judiciously, we could find useful constraints for expressions, a technique that's invaluable for inequality problem solving.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \(a, b, c\)), and operators (like +, -, *, and /). These expressions represent quantities that can vary; that is to say, they are not fixed. They form the backbone of algebra and allow us to describe relationships between variables and constants in equations and inequalities.

One of the most critical skills in solving algebra problems is the ability to manipulate these expressions effectively. This involves expanding them, factoring, and combining like terms. In our given problem, understanding how to expand squared terms and re-arrange them brings us closer to the solution. The key objective is to isolate the variables or terms we are solving for, such as \(ab + ac + bc\), to determine their possible values or describe their behavior within an interval.

Bounded Intervals

In mathematics, intervals are used to specify a range of values along a continuous scale, such as numbers on a line. When an interval is 'bounded', it means there are defined upper and lower limits to the values that can be taken within this range. These bounds can be inclusive or exclusive, which are respectively denoted by square \( [ ] \) and round brackets \( ( ) \).

Understanding bounded intervals is crucial when we are working with inequalities. They provide visual and conceptual ways to express constraints on variables. In the case of our exercise, we determined that the value of the expression \(ab + ac + bc\) must lie within the closed interval \( [-1/2,1] \). This means that the value of \(ab + ac + bc\) cannot be less than \( -1/2 \) nor greater than \(1\), capturing the essence of boundedness in the problem's context.