Exercises for Question Type 8: Determining components of a vector not acting or acting perpendicularly on another vector

Question 1

Skill question

Given $u = (1, 0, 2)$ , $v = (0, 1, 3)$ and $w = (2, 1, 1)$ , calculate the scalar triple product $u \cdot (v \times w)$ and hence determine the volume of the parallelepiped defined by $u, v$ , and $w$ .

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Question 2

Skill question

Using generic components $u = (u_{1}, u_{2}, u_{3})$ and $v = (v_{1}, v_{2}, v_{3})$ , prove that $u \times v = - v \times u$ .

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Question 3

Skill question

Find a unit vector perpendicular to both $u = (1, 2, 2)$ and $v = (2, - 1, 3)$ .

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Question 4

Skill question

Given $u = (2, - 1, 3)$ and $v = (0, 4, - 2)$ , compute $u \times v$ and verify that it is orthogonal to both $u$ and $v$ .

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Question 5

Skill question

Let $u = (1, 2, 3)$ , $v = (0, 1, 4)$ and $w = (2, 0, 1)$ . Decompose $w$ into a component perpendicular to the plane spanned by $u$ and $v$ and a component parallel to that plane.

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Question 6

Skill question

Show that for any vectors $u, v$ in $R^{3}$ , the squared magnitude of their cross product satisfies $∣ u \times v ∣^{2} = ∣ u ∣^{2} ∣ v ∣^{2} - (u \cdot v)^{2} .$

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Question 7

Skill question

If two vectors $u$ and $v$ have magnitudes $∣ u ∣ = 5$ , $∣ v ∣ = 7$ and the angle between them is $6 0^{\circ}$ , compute $∣ u \times v ∣$ .

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Question 8

Skill question

Consider the vectors $u = (a, b, c)$ and $v = (2 a, 2 b, 2 c)$ , where $a, b, c \in R$ .

Show that $u \times v = 0$ and conclude the relationship between $u$ and $v$ .

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Question 9

Skill question

Given the vectors $u = (1, 0, 1)$ , $v = (2, 1, 3)$ and $w = (3, 2, 5)$ , show that these vectors are coplanar.

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Question 10

Skill question

Give an explicit example of three vectors $u, v, w$ such that $(u \times v) \times w \neq = u \times (v \times w)$ , and compute both sides.

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Question 11

Skill question

This question requires students to calculate the cross product of two vectors and find its magnitude to determine the area of a parallelogram.

Given $u = (3, 1, 0)$ and $v = (1, 2, 2)$ , find the area of the parallelogram spanned by $u$ and $v$ .

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Question 12

Skill question

The question tests the properties of the cross product and dot product in $R^{3}$ . Specifically, it addresses the distributive property of the dot product and the geometric relationship between the cross product of two vectors and the vectors themselves.

Prove that for any nonzero vectors $u, v$ in $R^{3}$ , $(u \times v) \cdot (u + v) = 0.$

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IB Mathematics Applications & Interpretation Question Type 8: Determining Components of a Vector Not Acting or Acting Perpendicularly on Another Vector Exercises