The question asks to find the value of $c$ such that the angle between two vectors $\mathbf{a}$ and $\mathbf{b}$ is 60°.

The given vectors are:

$\mathbf{a} = i + cj + 2k \quad \text{and} \quad \mathbf{b} = -i + 2j - k$

We can calculate the angle $\theta$ between two vectors using the formula:

$\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$

Where:

• $\mathbf{a} \cdot \mathbf{b}$ is the dot product of vectors $\mathbf{a}$ and $\mathbf{b}$,
• $|\mathbf{a}|$ and $|\mathbf{b}|$ are the magnitudes of $\mathbf{a}$ and $\mathbf{b}$, respectively,
• $\theta$ is the angle between the vectors.

Since we are given that the angle between the vectors is 60°, we can substitute $\cos 60^\circ = \frac{1}{2}$ into the formula to find $c$.

Let's compute the necessary values and solve for $c$.

1. The dot product of $\mathbf{a}$ and $\mathbf{b}$:

$\mathbf{a} \cdot \mathbf{b} = (1)(-1) + (c)(2) + (2)(-1) = -1 + 2c - 2 = 2c - 3$

2. The magnitudes of $\mathbf{a}$ and $\mathbf{b}$:

$|\mathbf{a}| = \sqrt{1^2 + c^2 + 2^2} = \sqrt{1 + c^2 + 4} = \sqrt{c^2 + 5}$

$|\mathbf{b}| = \sqrt{(-1)^2 + 2^2 + (-1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6}$

Now substitute into the equation:

$\frac{2c - 3}{\sqrt{c^2 + 5} \times \sqrt{6}} = \frac{1}{2}$

We can now solve this equation to find the value of $c$.

Let's proceed with the solution. sympy import symbols, Eq, solve, sqrt

Define the variable

c = symbols('c')

Define the equation

equation = Eq((2*c - 3) / (sqrt(c**2 + 5) * sqrt(6)), 1/2)

Solve for c

solutions = solve(equation, c) solutionsThe solution to the equation is approximately $c \approx 4.67$.

Would you like further details or have any questions regarding this solution?

Here are some related questions:

1. How do we calculate the dot product of two vectors?
2. What is the formula to find the angle between two vectors?
3. How can we interpret the results of vector magnitudes?
4. What are some other methods to calculate the angle between vectors?
5. What are the conditions under which the angle between two vectors is 90°?

Tip: When solving vector problems involving angles, always ensure to check your units and double-check if you are using radians or degrees.