Let a and b be two unit vectors such that | a + b | = root 3. If c = a + 2b + 3 (a x b), then 2|c| is given by?

To solve the problem, we first need to analyze the given information about the unit vectors \( a \) and \( b \). Since both \( a \) and \( b \) are unit vectors, their magnitudes are 1, which will be important for our calculations. The condition \( |a + b| = \sqrt{3} \) gives us crucial insight into the relationship between these two vectors.

Understanding the Magnitude of the Sum of Vectors

We can use the formula for the magnitude of the sum of two vectors. For any two vectors \( a \) and \( b \), the magnitude of their sum is given by:

|a + b| = √(|a|² + |b|² + 2(a · b))

Since both \( a \) and \( b \) are unit vectors, we substitute |a| and |b| with 1:

|a + b| = √(1² + 1² + 2(a · b)) = √(2 + 2(a · b))

Given that \( |a + b| = \sqrt{3} \), we can set up the equation:

√(2 + 2(a · b)) = √3

Squaring both sides leads to:

2 + 2(a · b) = 3

From this, we simplify to find:

2(a · b) = 1

Thus, we have:

a · b = 1/2

This means that the angle between \( a \) and \( b \) is cos⁻¹(1/2), which corresponds to 60 degrees.

Calculating the Vector \( c \)

Now, let's move on to calculate the vector \( c = a + 2b + 3(a \times b) \). To find the magnitude of \( c \), we first need to find the components of \( c \). We will compute \( |c| \) step by step.

Breaking Down \( c \)

We already have the components \( a + 2b \). We will represent \( a \) and \( b \) in terms of their components:

• Let \( a = (1, 0) \) (without loss of generality).
• Then \( b \) can be represented as \( b = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \) due to the angle being 60 degrees.

Finding \( a + 2b \)

Now we calculate:

a + 2b = (1, 0) + 2\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) = (1 + 1, 0 + \sqrt{3}) = (2, \sqrt{3})

Calculating \( a \times b \)

The cross product \( a \times b \) in two dimensions can be represented as a scalar in the z-direction:

|a × b| = |a||b|sin(θ) = 1 * 1 * sin(60°) = \frac{\sqrt{3}}{2}

Thus, we can represent \( a \times b \) as:

a × b = \left(0, 0, \frac{\sqrt{3}}{2}\right)

Combining the Components

Now we can add \( 3(a \times b) \) to \( a + 2b \):

c = (2, \sqrt{3}) + 3(0, 0, \frac{\sqrt{3}}{2}) = (2, \sqrt{3}, \frac{3\sqrt{3}}{2})

Final Calculation of the Magnitude of \( c \)

To find \( |c| \), we use the formula for the magnitude of a vector in three dimensions:

|c| = √(x² + y² + z²)

Substituting the values we have:

|c| = √(2² + (\sqrt{3})² + \left(\frac{3\sqrt{3}}{2}\right)²)

Calculating each term:

• 2² = 4
• (√3)² = 3
• \(\left(\frac{3\sqrt{3}}{2}\right)² = \frac{27}{4}\)

Now summing these up:

|c| = √\left(4 + 3 + \frac{27}{4}\right) = √\left(\frac{16}{4} + \frac{12}{4} + \frac{27}{4}\right) = √\left(\frac{55}{4}\right)

Thus:

|c| = \frac{\sqrt{55}}{2}

Finding \( 2|c| \)

Finally, we need to calculate \( 2|c| \):

2|c| = 2 * \frac{\sqrt{55}}{2} = \sqrt{55}

In conclusion, the value of \( 2|c| \) is \( \sqrt{55} \).