Questions: Orthogonal or Perpendicular Vectors choose one 4 points - Which two vectors a and b are orthogonal (perpendicular)? a=(2,-1,3) and b=(0,3,1) a=(-1,-2,0) and b=(2,1,0) a=(0,-1,3) and b=(0,1,1)

Orthogonal or Perpendicular Vectors
choose one 4 points
- Which two vectors a and b are orthogonal (perpendicular)?

a=(2,-1,3) and b=(0,3,1)

a=(-1,-2,0) and b=(2,1,0)

a=(0,-1,3) and b=(0,1,1)

Transcript text: Orthogonal or Perpendicular Vectors choose one 4 points - Which two vectors $\vec{a}$ and $\vec{b}$ are orthogonal (perpendicular)? $\vec{a}=(2,-1,3)$ and $\vec{b}=(0,3,1)$ $\vec{a}=(-1,-2,0)$ and $\vec{b}=(2,1,0)$ $\vec{a}=(0,-1,3)$ and $\vec{b}=(0,1,1)$

Solution

Solution Steps

To determine if two vectors are orthogonal, we need to check if their dot product is zero. The dot product of two vectors $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$ is calculated as $a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3$. We will compute the dot product for each pair of vectors and check if it equals zero.

Step 1: Check Orthogonality of Vectors $\vec{a} = (2, -1, 3)$ and $\vec{b} = (0, 3, 1)$

Calculate the dot product: \[ \vec{a} \cdot \vec{b} = 2 \cdot 0 + (-1) \cdot 3 + 3 \cdot 1 = 0 - 3 + 3 = 0 \] Since the dot product is $0$, the vectors are orthogonal.

Step 2: Check Orthogonality of Vectors $\vec{a} = (-1, -2, 0)$ and $\vec{b} = (2, 1, 0)$

Calculate the dot product: \[ \vec{a} \cdot \vec{b} = (-1) \cdot 2 + (-2) \cdot 1 + 0 \cdot 0 = -2 - 2 + 0 = -4 \] Since the dot product is $-4$, the vectors are not orthogonal.

Step 3: Check Orthogonality of Vectors $\vec{a} = (0, -1, 3)$ and $\vec{b} = (0, 1, 1)$

Calculate the dot product: \[ \vec{a} \cdot \vec{b} = 0 \cdot 0 + (-1) \cdot 1 + 3 \cdot 1 = 0 - 1 + 3 = 2 \] Since the dot product is $2$, the vectors are not orthogonal.