Find a vector that is orthogonal to the given vector. 17. $$ \langle 10,-2\rangle $$ 18. $$ \langle-3,8\rangle $$

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To find a vector that is **orthogonal** (perpendicular) to a given vector in 2-dimensional space, you can use the following method:

Given a vector **v** = ⟨a, b⟩, an orthogonal vector **u** can be:

- **u** = ⟨b, -a⟩
  
This works because the dot product of **v** and **u** is zero:
$$ \mathbf{v} \cdot \mathbf{u} = a \cdot b + b \cdot (-a) = ab - ab = 0 $$

Now, let's apply this to the given vectors:

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### 17. Orthogonal to ⟨10, −2⟩

Given **v₁** = ⟨10, −2⟩,

An orthogonal vector **u₁** is:
$$ \mathbf{u}_1 = \langle -2, -10 \rangle $$

*Verification:*
$$ \mathbf{v}_1 \cdot \mathbf{u}_1 = 10 \times (-2) + (-2) \times (-10) = -20 + 20 = 0 $$

### 18. Orthogonal to ⟨−3, 8⟩

Given **v₂** = ⟨−3, 8⟩,

An orthogonal vector **u₂** is:
$$ \mathbf{u}_2 = \langle 8, 3 \rangle $$

*Verification:*
$$ \mathbf{v}_2 \cdot \mathbf{u}_2 = (-3) \times 8 + 8 \times 3 = -24 + 24 = 0 $$

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**Note:** There are infinitely many vectors orthogonal to a given vector. The vectors provided above are specific examples. Any scalar multiple of these vectors will also be orthogonal. For instance, scaling **u₁** by 2 gives ⟨-4, -20⟩, which is still orthogonal to **v₁**.
Orthogonal vectors to the given vectors are: 17. ⟨-2, -10⟩ 18. ⟨8, 3⟩

Find a vector that is orthogonal to the given vector. 17. \( \langle 10,-2\rangle \) 18. \( \langle-3,8\rangle \)

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