Given the vector v = <3, -6> and the product uv = 33, find u.
Understand the Problem
The question is presenting a vector and a product related to that vector (uv = 33). It appears to require solving for one of the variables or understanding the relationship between the vector and the scalar value provided.
Answer
$||\mathbf{u}|| = \frac{33}{||\mathbf{v}|| \cdot \cos(\theta)}$
Answer for screen readers
The relationship between the vectors can be expressed as $||\mathbf{u}|| = \frac{33}{||\mathbf{v}|| \cdot \cos(\theta)}$.
Steps to Solve
- Identify the vectors and the product
Assume we have two vectors $\mathbf{u}$ and $\mathbf{v}$. The inner product of these vectors is given by the equation: $$ \mathbf{u} \cdot \mathbf{v} = uv = 33 $$
- Express the inner product
The inner product (dot product) of two vectors $\mathbf{u}$ and $\mathbf{v}$ can be expressed as: $$ \mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| \cdot ||\mathbf{v}|| \cdot \cos(\theta) $$ where $||\mathbf{u}||$ and $||\mathbf{v}||$ are the magnitudes of the vectors and $\theta$ is the angle between them.
- Set up the equations
Given that the product of the vectors resolves to a scalar value (33), we can rearrange this equation to isolate either $||\mathbf{u}||$ or $||\mathbf{v}||$ or find a relationship between them.
If we wanted to solve for $||\mathbf{u}||$, we could write: $$ ||\mathbf{u}|| = \frac{33}{||\mathbf{v}|| \cdot \cos(\theta)} $$
- Determine any additional values (if needed)
To proceed further, you would need specific values for $||\mathbf{v}||$ or $\cos(\theta)$. If either is known, you can substitute and solve.
The relationship between the vectors can be expressed as $||\mathbf{u}|| = \frac{33}{||\mathbf{v}|| \cdot \cos(\theta)}$.
More Information
The inner product of vectors is a measure of how much one vector goes in the direction of another. If $\mathbf{u}$ and $\mathbf{v}$ are normalized (each has a magnitude of 1), their dot product simply equals the cosine of the angle between them.
Tips
- Assuming vectors always have the same magnitude or that both must be unit vectors without clarification.
- Neglecting the angle $\theta$ in the relationship; it must be considered when connecting magnitudes via the cosine function.
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