Math Problem Statement

Consider the a→=i→+4j→−k→,b→=−i→−4j→+k→,c→=4i→+j→+5k→d→=−i→−4j→+k→,g→=−i→−j→+4k→.Which pairs (if any) of these vectors are

(a) Are perpendicular?

(Enter none or a pair or list of pairs, e.g., if a→ is perpendicular to b→ and c→, enter (a,b),(a,c).)

(b) Are parallel?

(Enter none or a pair or list of pairs, e.g., if a→ is parallel to b→ and c→, enter (a,b),(a,c).)

(c) Have an angles less than π/2 between them?

(Enter none or a pair or list of pairs, e.g., if a→ is at an angle less than pi/2 from b→ and c→, enter (a,b),(a,c).)

(d) Have an angle of more than π/2 between them?

(Enter none or a pair or list of pairs, e.g., if a→ is at an angle greater than pi/2 from b→ and c→, enter (a,b),(a,c).)

Solution

Let's address each of the parts systematically.

We are given the following vectors:

a=i^+4j^−k^,b=−i^−4j^+k^,c=4i^+j^+5k^,d=−i^−4j^+k^,g=−i^−j^+4k^\mathbf{a} = \hat{i} + 4\hat{j} - \hat{k}, \quad \mathbf{b} = -\hat{i} - 4\hat{j} + \hat{k}, \quad \mathbf{c} = 4\hat{i} + \hat{j} + 5\hat{k}, \quad \mathbf{d} = -\hat{i} - 4\hat{j} + \hat{k}, \quad \mathbf{g} = -\hat{i} - \hat{j} + 4\hat{k}

(a) Are perpendicular?

Vectors are perpendicular if their dot product is zero, i.e., u⋅v=0\mathbf{u} \cdot \mathbf{v} = 0.

We calculate the dot product between each pair of vectors and check if it equals 0.

Dot products:

  1. a⋅b=(1)(−1)+(4)(−4)+(−1)(1)=−1−16−1=−18\mathbf{a} \cdot \mathbf{b} = (1)(-1) + (4)(-4) + (-1)(1) = -1 - 16 - 1 = -18

    • Not perpendicular.

  2. a⋅c=(1)(4)+(4)(1)+(−1)(5)=4+4−5=3\mathbf{a} \cdot \mathbf{c} = (1)(4) + (4)(1) + (-1)(5) = 4 + 4 - 5 = 3

    • Not perpendicular.

  3. a⋅d=(1)(−1)+(4)(−4)+(−1)(1)=−1−16−1=−18\mathbf{a} \cdot \mathbf{d} = (1)(-1) + (4)(-4) + (-1)(1) = -1 - 16 - 1 = -18

    • Not perpendicular.

  4. a⋅g=(1)(−1)+(4)(−1)+(−1)(4)=−1−4−4=−9\mathbf{a} \cdot \mathbf{g} = (1)(-1) + (4)(-1) + (-1)(4) = -1 - 4 - 4 = -9

    • Not perpendicular.

  5. b⋅c=(−1)(4)+(−4)(1)+(1)(5)=−4−4+5=−3\mathbf{b} \cdot \mathbf{c} = (-1)(4) + (-4)(1) + (1)(5) = -4 - 4 + 5 = -3

    • Not perpendicular.

  6. b⋅d=(−1)(−1)+(−4)(−4)+(1)(1)=1+16+1=18\mathbf{b} \cdot \mathbf{d} = (-1)(-1) + (-4)(-4) + (1)(1) = 1 + 16 + 1 = 18

    • Not perpendicular.

  7. b⋅g=(−1)(−1)+(−4)(−1)+(1)(4)=1+4+4=9\mathbf{b} \cdot \mathbf{g} = (-1)(-1) + (-4)(-1) + (1)(4) = 1 + 4 + 4 = 9

    • Not perpendicular.

  8. c⋅d=(4)(−1)+(1)(−4)+(5)(1)=−4−4+5=−3\mathbf{c} \cdot \mathbf{d} = (4)(-1) + (1)(-4) + (5)(1) = -4 - 4 + 5 = -3

    • Not perpendicular.

  9. c⋅g=(4)(−1)+(1)(−1)+(5)(4)=−4−1+20=15\mathbf{c} \cdot \mathbf{g} = (4)(-1) + (1)(-1) + (5)(4) = -4 - 1 + 20 = 15

    • Not perpendicular.

  10. (\mathbf{d} \cdot \mathbf{g} = (-1)(-1) + (-4)(-1) + (1)(4)

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Math Problem Analysis

Mathematical Concepts

Vector Dot Product
Vector Perpendicularity
Vector Parallelism
Angle Between Vectors

Formulas

Dot Product formula: u·v = u1v1 + u2v2 + u3v3
Perpendicular condition: u·v = 0
Parallel condition: u = k * v

Theorems

Condition for perpendicular vectors
Condition for parallel vectors

Suitable Grade Level

Grades 11-12

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