Quant ha de valer x per tal que els vectors ~u = (83, −2) i ~v = (x, 23)siguin perpendiculars

1. Break Down the Problem

To determine the value of $x$ such that the vectors $\mathbf{u} = (83, -2)$ and $\mathbf{v} = (x, 23)$ are perpendicular, we need to use the property that two vectors $\mathbf{a}$ and $\mathbf{b}$ are perpendicular if their dot product is zero.

2. Relevant Concepts

The dot product of two vectors $\mathbf{u} = (u_1, u_2)$ and $\mathbf{v} = (v_1, v_2)$ is defined as: $\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2$ Thus, for the vectors to be perpendicular: $\mathbf{u} \cdot \mathbf{v} = 0$

3. Analysis and Detail

Using the vectors given:

• $\mathbf{u} = (83, -2)$
• $\mathbf{v} = (x, 23)$

We can write the dot product: $\mathbf{u} \cdot \mathbf{v} = 83 \cdot x + (-2) \cdot 23$

Setting the dot product equal to zero: $83x - 46 = 0$

Now, let's solve for $x$: $83x = 46$ $x = \frac{46}{83}$

4. Verify and Summarize

To verify, substitute $x = \frac{46}{83}$ back into the dot product: $83 \left(\frac{46}{83}\right) - 46 = 46 - 46 = 0$ This confirms that our value of $x$ makes the vectors perpendicular.

Final Answer

The value of $x$ such that the vectors $\mathbf{u}$ and $\mathbf{v}$ are perpendicular is: $x = \frac{46}{83}$