Let's break down the problem into two parts:

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Part 1: Calculating the dot product of

$\vec{a}$ and $\vec{b}$

We are given the following dataLet's break down the problem into two parts:

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Part 1: Calculating the dot product of

$\vec{a}$ and $\vec{b}$

We are given the following

• $|\vec{a}| = 7$
• $|\vec{b}| = 9$
• The angle between $\vec{a}$ and $\vec{b}$, $v(\vec{a}, \vec{b}) = 13^\circ$

The dot product formula for two vectors is:

$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta)$

Where:

• $|\vec{a}|$ and $|\vec{b}|$ are the magnitudes of the vectors $\vec{a}$ and $\vec{b}$,
• $\theta$ is the angle between them, which is $13^\circ$ in this case.

Now let's calculate the dot product:

$\vec{a} \cdot \vec{b} = 7 \times 9 \times \cos(13^\circ)$

I'll compute this value.

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Part 2: Finding the value of $k$ for vectors $\vec{c} = \langle 20, 30 \rangle$ and $\vec{d} = \langle 50, k \rangle$

We are given:

• $\vec{c} = \langle 20, 30 \rangle$
• $\vec{d} = \langle 50, k \rangle$
• The angle between $\vec{c}$ and $\vec{d}$, $v(\vec{c}, \vec{d}) = 70^\circ$

The formula for the dot product of two vectors in terms of their components is:

$\vec{c} \cdot \vec{d} = 20 \times 50 + 30 \times k$

At the same time, using the dot product formula for the angle between two vectors:

$\vec{c} \cdot \vec{d} = |\vec{c}| |\vec{d}| \cos(70^\circ)$

Where:

• $|\vec{c}| = \sqrt{20^2 + 30^2} = \sqrt{400 + 900} = \sqrt{1300}$
• $|\vec{d}| = \sqrt{50^2 + k^2} = \sqrt{2500 + k^2}$

Now we have two expressions for the dot product, and we can solve for $k$ by equating them.

Let's calculate $k$.

Part 1: Dot Product of $\vec{a} \cdot \vec{b}$

The dot product of the vectors $\vec{a}$ and $\vec{b}$ is approximately:

$\vec{a} \cdot \vec{b} = 61.39$

Part 2: Finding $k$

The value of $k$ that satisfies the condition where the angle between $\vec{c} = \langle 20, 30 \rangle$ and $\vec{d} = \langle 50, k \rangle$ is $70^\circ$ is approximately:

$k \approx -12.18$

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Would you like a detailed explanation or sketch of the vectors?

5 Related Questions:

1. How is the dot product of two vectors related to the angle between them?
2. How can the magnitude of a vector be computed from its components?
3. What does it mean if the dot product of two vectors is zero?
4. How does changing the angle between two vectors affect their dot product?
5. Can you solve for $k$ if the angle between vectors is $90^\circ$?

Tip:

The dot product is a powerful tool for calculating angles and projections between vectors!