Question

For vector \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) if each vector forms an angle \((\pi / 3)\) with remaining two vectors and \(|\underline{a}|=1,|\underline{b}|=2,|\underline{c}|=3\), then \(|\underline{a}+\underline{b}+\underline{c}|=\) (A) \(\sqrt{17}\) (B) 0 (C) 5 (D) \(\sqrt{5}\)

Short answer

The magnitude of the sum of the three vectors is 5, which corresponds to option (C).

Step-by-step solution

Compute the dot products of pairs of vectors

Using the formula for the dot product of two vectors given their magnitudes and the angle between them, we find the dot products for each pair of vectors: \(\underline{a} \cdot \underline{b} = |\underline{a}| \ |\underline{b}| \cos(\pi / 3) = (1)(2)\cos(\pi / 3) = 1\) \(\underline{a} \cdot \underline{c} = |\underline{a}| \ |\underline{c}| \cos(\pi / 3) = (1)(3) \cos(\pi / 3) = \frac{3}{2}\) \(\underline{b} \cdot \underline{c} = |\underline{b}| \ |\underline{c}| \cos(\pi / 3) = (2)(3) \cos(\pi / 3) = 3\)

Apply dot product properties to find the square of the magnitude

We know that the square of the magnitude of a vector sum is equal to the sum of the squares of their magnitudes plus 2 times the sum of the products of their components: \(|\underline{a} + \underline{b} + \underline{c}|^2 = |\underline{a}|^2 + |\underline{b}|^2 + |\underline{c}|^2 + 2(\underline{a}\cdot\underline{b}+\underline{a}\cdot\underline{c}+\underline{b}\cdot\underline{c})\)

Substitute the values computed in steps 1 and 2

Plugging the values from step 1 into the equation above, we get: \(|\underline{a} + \underline{b} + \underline{c}|^2 = 1^2 + 2^2 + 3^2 + 2(1 + \frac{3}{2} + 3) = 1 + 4 + 9 + 2(\frac{11}{2}) = 14 + 11 = 25\)

Compute the magnitude

Now that we have the square of the magnitude, we can find the magnitude by taking the square root: \(|\underline{a} + \underline{b} + \underline{c}| = \sqrt{25} = 5\) So, the magnitude of the sum of the three vectors is 5, which corresponds to option (C).

Key concepts

Dot Product of Vectors

The dot product, also known as the scalar product, is a way to multiply two vectors to get a scalar value. This operation reveals how much of one vector goes in the direction of another. To calculate the dot product of two vectors, you need their magnitudes and the angle between them. The formula for the dot product of vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by:

• \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) \)

Here, \( |\mathbf{a}| \) and \( |\mathbf{b}| \) are the magnitudes of vectors \( \mathbf{a} \) and \( \mathbf{b} \), and \( \theta \) is the angle between them.
In the original exercise, the dot products of pairwise vectors were calculated: \( \underline{a} \cdot \underline{b} = 1 \), \( \underline{a} \cdot \underline{c} = \frac{3}{2} \), and \( \underline{b} \cdot \underline{c} = 3 \).
These values are crucial because they account for the interaction between each vector within the vector set, ultimately assisting in computing the magnitude of the sum of these vectors.

Vector Magnitude

Magnitude of a vector is a measure of its length. For any vector \( \mathbf{v} = (x, y, z) \), the magnitude is computed as:

• \( |\mathbf{v}| = \sqrt{x^2 + y^2 + z^2} \)

In the exercise, it was given that the magnitudes of vectors were \( |\underline{a}| = 1 \), \( |\underline{b}| = 2 \), and \( |\underline{c}| = 3 \). These values are fundamental because when you need to find the magnitude of the vector sum \( |\underline{a} + \underline{b} + \underline{c}| \), they are part of the equation. The magnitude of a vector sum can be found using the relation:

• \( |\mathbf{a} + \mathbf{b} + \mathbf{c}|^2 = |\mathbf{a}|^2 + |\mathbf{b}|^2 + |\mathbf{c}|^2 + 2(\underline{a}\cdot\underline{b} + \underline{a}\cdot\underline{c} + \underline{b}\cdot\underline{c})\)

In the problem, after substituting the values, the magnitude squared was found to be 25, leading to a final magnitude of 5.

Angle Between Vectors

The angle between vectors plays a crucial role in calculations involving vectors, particularly in the dot product. The angle \( \theta \) between two vectors \( \mathbf{a} \) and \( \mathbf{b} \) relates to their dot product and quantifies their directional relationship.

• If \( \theta = 0 \), the vectors are aligned in the same direction.
• If \( \theta = \frac{\pi}{2} \), the vectors are perpendicular.
• If \( \theta = \pi \), the vectors point in the opposite direction.

In the given exercise, each vector formed an angle \( \frac{\pi}{3} \) with the other two vectors. The cosine of \( \frac{\pi}{3} \) is \( \frac{1}{2} \), a key value used to compute individual dot products between vectors. Understanding the angle between vectors helps to identify how much influence one vector has over another, which in turn impacts the calculation of the vector sum results.