Question

1. determine if the two given vectors are orthogonal. if they are not orthogonal, then determine if the angle between them is acute or obtuse. 1. ⟨2,3⟩ and ⟨3,−2⟩. 2. ⟨2,3⟩ and ⟨−3,−2⟩. 3. ⟨2,3⟩ and ⟨3,2⟩. 4. ⟨4,4,−2⟩ and ⟨3,−2,2⟩. 5. ⟨1,0,2⟩ and ⟨−4,−1,3⟩. 6. ⟨1,1,2⟩ and ⟨4,4,−2⟩.

Explanation:

Step1: Recall dot product rule

For vectors $\langle a_1,a_2,...,a_n
angle$ and $\langle b_1,b_2,...,b_n
angle$, dot product is $\sum_{i=1}^n a_i b_i$.

• If dot product = 0: vectors are orthogonal
• If dot product > 0: angle is acute
• If dot product < 0: angle is obtuse

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Subproblem 1: $\langle 2,3

angle$ and $\langle 3,-2
angle$

Step1: Calculate dot product

$\langle 2,3
angle \cdot \langle 3,-2
angle = (2)(3) + (3)(-2)$
$= 6 - 6 = 0$

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Subproblem 2: $\langle 2,3

angle$ and $\langle -3,-2
angle$

Step1: Calculate dot product

$\langle 2,3
angle \cdot \langle -3,-2
angle = (2)(-3) + (3)(-2)$
$= -6 - 6 = -12$

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Subproblem 3: $\langle 2,3

angle$ and $\langle 3,2
angle$

Step1: Calculate dot product

$\langle 2,3
angle \cdot \langle 3,2
angle = (2)(3) + (3)(2)$
$= 6 + 6 = 12$

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Subproblem 4: $\langle 4,4,-2

angle$ and $\langle 3,-2,2
angle$

Step1: Calculate dot product

$\langle 4,4,-2
angle \cdot \langle 3,-2,2
angle = (4)(3) + (4)(-2) + (-2)(2)$
$= 12 - 8 - 4 = 0$

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Subproblem 5: $\langle 1,0,2

angle$ and $\langle -4,-1,3
angle$

Step1: Calculate dot product

$\langle 1,0,2
angle \cdot \langle -4,-1,3
angle = (1)(-4) + (0)(-1) + (2)(3)$
$= -4 + 0 + 6 = 2$

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Subproblem 6: $\langle 1,1,2

angle$ and $\langle 4,4,-2
angle$

Step1: Calculate dot product

$\langle 1,1,2
angle \cdot \langle 4,4,-2
angle = (1)(4) + (1)(4) + (2)(-2)$
$= 4 + 4 - 4 = 4$

Answer:

1. The vectors are orthogonal.
2. The angle between the vectors is obtuse.
3. The angle between the vectors is acute.
4. The vectors are orthogonal.
5. The angle between the vectors is acute.
6. The angle between the vectors is acute.