The vector product(2)

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26 Terms

1
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What is the formula for the vector product a×b?

θ= the angle between the vectors

n^= the unit vector perpendicular to both a and b.

<p>θ= the angle between the vectors</p><p><strong>n^</strong>= the unit vector perpendicular to both <strong>a</strong> and <strong>b</strong>.</p>
2
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How do you show that 3 vectors are coplanar(lie in the same plane)?

The result of the triple scalar product will=0 if they’re coplanar

<p>The result of the triple scalar product will=0 if they’re coplanar </p>
3
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What is the result of the vector (cross) product a×b?

The result of the vector product is a vector that has:

  • Magnitude ∣a∣∣b∣sin⁡θ|, corresponding to the area of the parallelogram formed by a and b

  • Direction perpendicular to both a and b, determined by the right-hand rule.

<p>The result of the vector product is a <strong>vector</strong> that has:</p><p></p><ul><li><p>Magnitude ∣<strong>a</strong>∣∣<strong>b</strong>∣sin⁡θ|, corresponding to the area of the parallelogram formed by <strong>a </strong>and <strong>b</strong></p><p></p></li><li><p>Direction perpendicular to both <strong>a</strong> and<strong> b</strong>, determined by the right-hand rule.</p></li></ul><p></p>
4
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How is the direction of the result of a vector product determined?

  • The direction is perpendicular to both vectors a and b

  • It is determined using the right-hand rule.

<ul><li><p>The direction is perpendicular to both vectors<strong> a</strong> and<strong> b</strong></p><p></p></li><li><p>It is determined using the <strong>right-hand rule</strong>.</p></li></ul><p></p>
5
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What does the magnitude of the vector product a×b represent?

The magnitude represents the area of the parallelogram formed by vectors a and b.

<p>The magnitude represents the area of the parallelogram formed by vectors <strong>a</strong> and <strong>b</strong>.</p>
6
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What is the distributive property of the vector product?

<p></p>
7
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What are the properties of the vector product?

  • Distributive

  • Anticommutative

  • Non-associative

8
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What is the anticommutative property of the vector product?

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9
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Is the vector product associative?

No, the vector product is non-associative:

<p>No, the vector product is <strong>non-associative</strong>:</p>
10
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What is the vector product of a vector with itself?

The vector product of a vector with itself is zero since the angle θ=0

<p>The vector product of a vector with itself is zero since the angle θ=0</p>
11
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If a×b=0, what can we conclude about the relationship between a and b?

a is parallel or antiparallel to b

12
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What is the result of the vector product for the Cartesian basis vectors?

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13
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How do you compute the vector product of two vectors in Cartesian components?

<p></p>
14
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How can you express the vector product of two vectors in a more compact form?

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15
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What is the scalar triple product of three vectors a, b, c?

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16
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What type of quantity does the scalar triple product result in?

The scalar triple product results in a scalar quantity.

17
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How can the scalar triple product be interpreted geometrically?

The scalar triple product represents the volume of a parallelepiped whose edges are given by the vectors a, b and c

<p>The scalar triple product represents the <strong>volume of a parallelepiped</strong> whose edges are given by the vectors <strong>a, b</strong> and<strong> c</strong></p>
18
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What does it mean if the scalar triple product (a×b)⋅c =0?

If the scalar triple product equals zero, then the vectors a, b and c are coplanar.

19
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Is the scalar triple product commutative?

Yes, the scalar triple product is cyclically commutative:

<p>Yes, the scalar triple product is <strong>cyclically commutative</strong>:</p>
20
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What happens if there are duplicated vectors in the scalar triple product?

If any vector is duplicated, the scalar triple product is zero

<p>If any vector is duplicated, the scalar triple product is zero</p>
21
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How do you compute the scalar triple product using the components in the Cartesian basis set?

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22
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How can the scalar triple product be expressed in determinant form?

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23
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<p>What is the vector triple product of three vectors<strong> a, b</strong> and <strong>c</strong>?</p>

What is the vector triple product of three vectors a, b and c?

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24
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What is the geometric property of the resultant vector in a vector triple product?

  • The resultant vector is perpendicular to both a and b×c

  • Additionally, b×c is perpendicular to both b and c.

<ul><li><p>The resultant vector is <strong>perpendicular</strong> to both <strong>a</strong> and <strong>b</strong>×<strong>c</strong> </p></li></ul><p></p><ul><li><p>Additionally, <strong>b</strong>×<strong>c</strong> is perpendicular to both <strong>b</strong> and <strong>c</strong>.</p></li></ul><p></p>
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Where does the resultant vector of the vector triple product lie?

The resultant vector lies in the plane containing b and c, and is a linear combination of b and c.

<p>The resultant vector lies in the plane containing<strong> b </strong>and <strong>c</strong>, and is a linear combination of <strong>b </strong>and <strong>c</strong>.</p>
26
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How can the vector triple product be expressed in a simplified form?

This is known as the "bac cab" rule.

<p>This is known as the "bac cab" rule.</p>