Question

If angle between two units vectors \(\underline{a}\) and \(\underline{b}\) is \(\theta\), then \(\sin (\theta / 2)=\) (A) \(|\underline{a}+\underline{b}|\) (B) \((1 / 2)|\underline{a}-\underline{b}|\) (C) \(|\underline{a}-\underline{b}|\) (D) \((1 / 2)|\underline{a}+\underline{b}|\)

Short answer

\(\sin(\theta / 2) = \frac{1}{2} \sqrt{2(1 - \underline{a} . \underline{b})}\)

Step-by-step solution

Dot product formula and unit vector properties

Recall the formula for the dot product \(\underline{a} . \underline{b}\) of two vectors \(\underline{a}\) and \(\underline{b}\) in terms of their magnitudes and the angle between them: \(\underline{a} . \underline{b} = |\underline{a}| |\underline{b}| \cos{\theta}\) Also, recall that the magnitude of a unit vector is 1. Thus, for our unit vectors \(\underline{a}\) and \(\underline{b}\), we have: \(|\underline{a}| = |\underline{b}| = 1\)

Using the dot product formula

Using the dot product formula and the properties of unit vectors, we have: \(\underline{a} . \underline{b} = 1 * 1 * \cos{\theta} = \cos{\theta}\)

Using the half-angle formula for sine

Recall the half-angle formula for sine: \(\sin(\theta / 2) = \sqrt{\frac{1 - \cos{\theta}}{2}}\) Using our expression for \(\cos{\theta}\) from Step 2, we have: \(\sin(\theta / 2) = \sqrt{\frac{1 - \underline{a} . \underline{b}}{2}}\)

Rearranging the expression for sine of half-angle

Let's rearrange the expression for \(\sin(\theta / 2)\): \(\sin(\theta / 2) = \frac{1}{2} \sqrt{2(1 - \underline{a} . \underline{b})}\)

Comparing with the given options

Now we compare our expression for \(\sin(\theta / 2)\) with the given options. (A) \(|\underline{a}+\underline{b}|\) (B) \((1 / 2)|\underline{a}-\underline{b}|\) (C) \(|\underline{a}-\underline{b}|\) (D) \((1 / 2)|\underline{a}+\underline{b}|\) None of these options match with our expression for \(\sin(\theta / 2)\). Therefore, the given options do not contain the correct answer. Note: This exercise might contain an error, and the correct option should have been written as \(\frac{1}{2} \sqrt{2(1 - \underline{a} . \underline{b})}\).

Key concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It takes two equally-sized vectors and returns a single scalar quantity.

Mathematically, the dot product \(\underline{a} \cdot \underline{b}\) is calculated by multiplying corresponding components of the two vectors and then summing those products. The dot product formula in relation to the angle between the vectors is:
\[\underline{a} \cdot \underline{b} = |\underline{a}| |\underline{b}| \cos{\theta}\]
Where \(\cos{\theta}\) is the cosine of the angle \(\theta\) between vectors \(\underline{a}\) and \(\underline{b}\), and \(\underline{a}\) and \(\underline{b}\) represent the magnitudes, or lengths, of vectors \(\underline{a}\) and \(\underline{b}\) respectively.

For unit vectors, the magnitude is always 1. As a result, the dot product for unit vectors simplifies to just the cosine of the angle between them:
\[\underline{a} \cdot \underline{b} = \cos{\theta}\]
This relationship is particularly useful in problems involving angles and lengths of vectors, such as determining the angle between two vectors or projecting one vector onto another.

Vector Magnitude

Vector magnitude is the length of a vector and is a measure of its size. The magnitude is always a non-negative scalar and can be thought of as the distance from the origin to the point represented by the vector in a coordinate space.

To find the magnitude of a vector \(\underline{v} = (v_x, v_y, v_z)\), you can use the formula:
\[|\underline{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}\]
This is analogous to finding the hypotenuse of a right triangle through the Pythagorean theorem in three-dimensional space.

For unit vectors, the magnitude is defined to be 1. This makes computations involving unit vectors more straightforward since their magnitude does not need to be calculated—they are inherently normalized to length 1. The concept of vector magnitude is used in conjunction with the dot product when expressing the relationship between vectors and angles, as seen in the problem from the textbook.

Half-Angle Formulas

Half-angle formulas are trigonometric identities that relate the sine, cosine, or tangent of half of an angle to the square root of expressions involving the sine, cosine, or tangent of the full angle. They are useful in various areas of mathematics and physics where angles and their trigonometric functions play a role.

One such formula for sine is:
\[\sin(\theta / 2) = \sqrt{\frac{1 - \cos{\theta}}{2}}\]
These formulas are derived from the double-angle formulas that express trigonometric functions of double angles in terms of single angles. To apply a half-angle formula, one might first find \(\cos{\theta}\) or \(\sin{\theta}\) using standard trigonometric functions or vector relationships, like the dot product, and then use the half-angle formulas to find the trigonometric values of \(\theta / 2\).

Indeed, the half-angle formulas were essential in the textbook problem to relate the dot product of two unit vectors to the sine of half the angle between them. An understanding of these formulas is vital for solving complex problems involving trigonometric functions and their properties.