Question

For vector a and \(\underline{b},|\underline{a}+\underline{b}|<|\underline{a}-\underline{b}|\), then the angle between \(\underline{a}\) and \(\underline{b}\) is (A) obtuse (B) Acute (C) Right (D) supplementary

Short answer

The angle between the vectors \(\underline{a}\) and \(\underline{b}\) is obtuse.

Step-by-step solution

Write down given inequality

We are given the inequality: \[|\underline{a} + \underline{b}| < |\underline{a} - \underline{b}|\]

Use dot product properties

Recall that for any two vectors \(\underline{v}\) and \(\underline{w}\), \(|\underline{v}|^2 = \underline{v} \cdot \underline{v}\). Now let's square both sides of the inequality: \[|(\underline{a}+\underline{b})|^2<|(\underline{a}-\underline{b})|^2\] Using the properties of dot products and expanding on both sides of the inequality, we have: \[(\underline{a}+\underline{b}) \cdot (\underline{a}+\underline{b})<(\underline{a}-\underline{b})\cdot(\underline{a}-\underline{b})\]

Apply the distributive property of dot products

Applying the distributive property of dot products on both sides, we get: \[\underline{a}\cdot\underline{a}+2(\underline{a}\cdot\underline{b})+\underline{b}\cdot\underline{b}<\underline{a}\cdot\underline{a}-2(\underline{a}\cdot\underline{b})+\underline{b}\cdot\underline{b}\]

Cancelling and simplifying terms

Now let's cancel and simplify the terms in the expression: \[4(\underline{a}\cdot\underline{b})<0\] Divide both sides by 4: \[\underline{a} \cdot \underline{b} < 0\]

Using the relationship between dot products and angles

The dot product of two vectors \(\underline{a}\) and \(\underline{b}\) can be expressed in terms of their magnitudes and the angle between them: \[\underline{a}\cdot\underline{b}=|\underline{a}||\underline{b}|\cos\theta\] Since \(\underline{a}\cdot\underline{b}<0\), this implies that \(\cos\theta<0\). We know that cosine is negative in the second and third quadrants (from \(90^\circ\) to \(270^\circ\)). This means that the angle between these two vectors is an obtuse angle. The correct choice is: (A) obtuse

Key concepts

Vector Inequality

The concept of vector inequality serves as an important foundation in vector algebra. In vector algebra, an inequality involving vectors, such as \(|\underline{a} + \underline{b}| < |\underline{a} - \underline{b}|\), indicates a comparison between the magnitudes of two resultant vectors. Here, these magnitudes are calculated by adding and subtracting the vectors \(\underline{a}\) and \(\underline{b}\) respectively.
This type of inequality might suggest how the direction and relative positions of vectors affect their combined lengths. A smaller magnitude for \(|\underline{a} + \underline{b}|\) compared to \(|\underline{a} - \underline{b}|\) suggests that the resulting vector from addition is shorter than that from subtraction. This can lead to insights about how the vectors are aligned with respect to one another.

Dot Product Properties

The dot product is a crucial concept in vector algebra, combining both magnitude and direction. It is represented mathematically as \(\underline{a} \cdot \underline{b} = |\underline{a}||\underline{b}|\cos\theta\).
The dot product has several properties:

• Distributive: \(\underline{a} \cdot (\underline{b} + \underline{c}) = \underline{a} \cdot \underline{b} + \underline{a} \cdot \underline{c}\)
• Commutative: \(\underline{a} \cdot \underline{b} = \underline{b} \cdot \underline{a}\)
• Associative with scalars: \(k(\underline{a} \cdot \underline{b}) = (k\underline{a}) \cdot \underline{b}\) where \(k\) is a scalar

Understanding these properties is essential, as they allow simplification and manipulation of vector expressions, especially when dealing with dot product equations or inequalities.

Angle Between Vectors

The angle between two vectors is fundamental in identifying their directional relationship. This angle, \(\theta\), can be deduced using the formula for the dot product: \(\underline{a} \cdot \underline{b} = |\underline{a}||\underline{b}|\cos\theta\).
From this equation, if the dot product is known, the cosine of the angle can be calculated as \(\cos\theta = \frac{\underline{a} \cdot \underline{b}}{|\underline{a}||\underline{b}|}\). Here are a few things to consider based on the value of \(\cos\theta\):

• If \(\cos\theta > 0\), the angle is acute (less than \(90^\circ\)).
• If \(\cos\theta = 0\), the vectors are perpendicular.
• If \(\cos\theta < 0\), the angle is obtuse (more than \(90^\circ\)).

This mathematical relationship helps in determining how vectors relate to each other in a spatial context.

Obtuse Angle Determination

Determining whether the angle between two vectors is obtuse involves interpreting the sign of the dot product. If the dot product is negative, as seen in the inequality \(\underline{a} \cdot \underline{b} < 0\), the angle \(\theta\) is obtuse. This occurs because \(\cos\theta < 0\) only when \(\theta\) falls between \(90^\circ\) and \(180^\circ\).
This means when vectors are positioned such that they point in generally opposite directions relative to one another, the product's cosine value becomes negative.
The concept is critical for understanding the directional orientations of vectors in physics and engineering, where vectors representing forces, velocities, or other quantities often need analysis for angles and directional influence.