Questions: Which pair of vectors is orthogonal? ⟨3,-5⟩ and ⟨6,-10⟩ ⟨2,-3⟩ and ⟨9,-6⟩ ⟨-2,3⟩ and ⟨-6,-4⟩ ⟨-5,4⟩ and ⟨-4,-2⟩

Transcript text: Which pair of vectors is orthogonal? $\langle 3,-5\rangle$ and $\langle 6,-10\rangle$ $\langle 2,-3\rangle$ and $\langle 9,-6\rangle$ $\langle-2,3\rangle$ and $\langle-6,-4\rangle$ $\langle-5,4\rangle$ and $\langle-4,-2\rangle$

Solution

Solution Steps

To determine which pair of vectors is orthogonal, we need to calculate the dot product of each pair. Two vectors are orthogonal if their dot product is zero. We will compute the dot product for each pair and check if it equals zero.

Step 1: Define Orthogonality

Two vectors $\mathbf{a} = \langle a_1, a_2 \rangle$ and $\mathbf{b} = \langle b_1, b_2 \rangle$ are orthogonal if their dot product is zero. The dot product is calculated as: \[ \mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2 \]

Step 2: Calculate Dot Products

We calculate the dot product for each pair of vectors:

For $\langle 3, -5 \rangle$ and $\langle 6, -10 \rangle$: \[ 3 \cdot 6 + (-5) \cdot (-10) = 18 + 50 = 68 \]
For $\langle 2, -3 \rangle$ and $\langle 9, -6 \rangle$: \[ 2 \cdot 9 + (-3) \cdot (-6) = 18 + 18 = 36 \]
For $\langle -2, 3 \rangle$ and $\langle -6, -4 \rangle$: \[ (-2) \cdot (-6) + 3 \cdot (-4) = 12 - 12 = 0 \]
For $\langle -5, 4 \rangle$ and $\langle -4, -2 \rangle$: \[ (-5) \cdot (-4) + 4 \cdot (-2) = 20 - 8 = 12 \]

Step 3: Identify Orthogonal Pair

The pair of vectors that has a dot product of zero is $\langle -2, 3 \rangle$ and $\langle -6, -4 \rangle$.