84.1 Cross Product Concept Overview

CROSS PRODUCT | CONCEPT OVERVIEW The topic of the CROSS PRODUCT can be referenced on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.

CONCEPT INTRO: The CROSS PRODUCT is one of two OPERATIONS that can be deployed using VECTOR MULTIPLICATION, the other operation being the DOT PRODUCT. In it’s most fundamental form, the CROSS PRODUCT takes two vectors, runs them through a standard process, and creates a third vector that is ORTHOGONAL (or PERPENDICULAR) to each of the original vectors. Whereas the DOT PRODUCT returns to us a SCALAR VALUE, the CROSS PRODUCT will result in a VECTOR with a defined LENGTH (or MAGNITUDE) and DIRECTION. The STANDARD NOMENCLATURE of CROSS PRODUCT is: 𝐴×𝐵 Don’t confuse the CROSS PRODUCT OPERATOR (×) with the x-style multiplication operator (𝑥) that we use everywhere else in MATHEMATICS. These are DISTINCT OPERATIONS.



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The GENERAL FORMULA for the CROSS PRODUCT can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Given the two vectors defined as: 𝐴 =< 𝑎! , 𝑎! , 𝑎! > 𝐵 =< 𝑏! , 𝑏! , 𝑏! > Having an angle 𝜃 between them, where 𝜃 is between 0 𝑎𝑛𝑑 𝜋, the CROSS PRODUCT is found using the formula: 𝐴 × 𝐵 =< 𝑎! 𝑏! − 𝑎! 𝑏! , 𝑎! 𝑏! − 𝑎! 𝑏! , 𝑎! 𝑏! − 𝑎! 𝑏! >= 𝐵 𝐴 𝑠𝑖𝑛𝜃 The CROSS PRODUCT of VECTORS 𝐴 and 𝐵 is a new vector PERPENDICULAR TO BOTH 𝐴 and 𝐵 and has a MAGNITUDE EQUAL to the AREA of the PARALLELOGRAM generated from 𝐴 to 𝐵, such that:



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The AREA for a PARALLELOGRAM is calculated as: 𝑤𝑖𝑑𝑡ℎ 𝑥 ℎ𝑒𝑖𝑔ℎ𝑡 Where the height is represented by the term: ℎ = 𝐵 𝑠𝑖𝑛𝜃 Which gives us a GENERAL FORMULA in relationship with our two vectors as: 𝐴 × 𝐵 = 𝐴𝑅𝐸𝐴 𝑂𝐹 𝑃𝐴𝑅𝐴𝐿𝐿𝐸𝐿𝑂𝐺𝑅𝐴𝑀 = 𝐴ℎ = 𝐴 𝐵 𝑠𝑖𝑛𝜃 To reiterate, unlike the DOT PRODUCT, which returns a SCALAR QUANTITY, the CROSS PRODUCT always returns a NEW VECTOR. The FORMULA for the ANTI-COMMUTATIVE PROPERTY of CROSS PRODUCTS can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The ANTI-COMMUTATIVE PROPERTY of CROSS PRODUCTS is generally stated using the expression: 𝐴 × 𝐵 = −𝐵 × 𝐴 The FORMULA for the DISTRIBUTIVE PROPERTY of CROSS PRODUCTS can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.

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The DISTRIBUTIVE PROPERTY of CROSS PRODUCTS is generally stated using the expression: 𝐴× 𝐵+𝐶 = 𝐴×𝐵+𝐴×𝐶 The FORMULAS for the CROSS PRODUCT of TWO PARALLEL VECTORS can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The CROSS PRODUCT produces a VECTOR that is ORTHOGONAL to the two original vectors, having a DIRECTION and a MAGNITUDE. There is one unique case where the MAGNITUDE of the VECTOR resulting from the CROSS PRODUCT will be ZERO and that is when the vectors being used in the operation ore ORTHAGONAL (or PARALLEL) with one another. This can be best represented using the expression: 𝑖×𝑖 =𝑗×𝑗 =𝑘×𝑘 =0 If 𝐴 × 𝐵 = 0, then either 𝐴 = 0, 𝐵 = 0, or 𝐴 is parallel to 𝐵. This is true since two vectors are parallel if and only if the angle between them is 0 degrees (or 180 degrees).



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The GENERAL FORMULA for the SENSE of a CROSS PRODUCT can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The SENSE (or DIRECTION) of the CROSS PRODUCT is determined using the RIGHT HAND RULE, and can be written in formulaic terms as: 𝐴 × 𝐵 = 𝐴 𝐵 𝑛 𝑠𝑖𝑛𝜃 Where: • 𝑛 is a UNIT VECTOR perpendicular to the plane of vectors 𝐴 and 𝐵 The DIRECTION of 𝑛 follows the RIGHT HAND RULE, which states that if given: 𝐴×𝐵 To determine the SENSE of our resulting VECTOR, we can point our thumb in the direction of VECTOR A, the first vector in the CROSS PRODUCT, and then point our fingers in the direction of VECTOR B. The result will have our palm facing in the direction of vector 𝑛. The FORMULAS for the SENSE of a CROSS PRODUCT in the CARTESIAN COORDINATE SYSTEM can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.

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The following relationships are used to define the SENSE of a CROSS PRODUCT that falls within a THREE DIMENSIONAL CARTESIAN COORDINATE SYSTEM, these to relationships formulate the expressions: 𝑖×𝑗 =𝑘 =𝑗×𝑖 𝑗 × 𝑘 = 𝑖 = −𝑘 × 𝑗 𝑘 × 𝑖 = 𝑗 = −𝑖 × 𝑘 These relationships can help you quickly establish the direction of a resulting vector from a CROSS PRODUCT. However, if you choose, you can define it just as quickly using the RIGHT HAND RULE. It is important to be cognizant of the hand you use to during the exam. We don’t say this in jest, but just based on our natural tendencies as right handed people to use our left hand during the exam. We do this because we are busy using the right hand to write, type, or select answers on the screen. Make sure to use your RIGHT HAND, as the left hand will result in a wrong conclusion. The GENERAL FORMULA for the CROSS PRODUCT in MATRIX FORM can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.



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We are given the GENERAL FORMULA of the CROSS PRODUCT in pure formulaic terms. However, we can set up and evaluate the CROSS PRODUCT of two vector using a matrix form…which in fact, will result in the same exact term by term formula we stated earlier. Illustrating the CROSS PRODUCT in a 3𝑥3 matrix, we get: 𝑖 𝐴 × 𝐵 = 𝑎! 𝑏!

𝑗 𝑎! 𝑏!

𝑘 𝑎! = −𝐵 × 𝐴 𝑏!

The first line of the determinant always contains the unit vectors in the direction of each of the Cartesian axes. The second line is always the scalar coefficients of each of the unit vectors for the first vector listed in the cross product. The third line is always the scalar coefficients of each of the unit vectors for the second vector listed in the cross product. We can deploy our knowledge of the METHOD OF COFACTORS to determine the cross product formula from this point. This method can generally be illustrated as: 𝑎! 𝐴×𝐵 = 𝑏 !



𝑎! 𝑎! 𝑖 − 𝑏! 𝑏! Made with

𝑎! 𝑎! 𝑗 + 𝑏! 𝑏!

𝑎! 𝑏! 𝑘

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Where: 𝑎 𝑐

𝑏 = 𝑎𝑑 − 𝑏𝑐 𝑑

It is important to note that in the METHOD OF COFACTORS, the TERMS ALTERNATE IN SIGN and each term is simply a 2𝑥2 matrix multiplied by a SCALAR.



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CROSS PRODUCT | CONCEPT EXAMPLES The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material. Given the two vectors 𝐴 =< 2, 1, −1 > and 𝐵 =< −3, 4, 1 >, the cross product, 𝐴 × 𝐵, results in the new vector: A. 5𝑖 + 11𝑘 B. 15𝑖 − 𝑗 + 11𝑘 C. 5𝑖 − 𝑗 + 21𝑘 D. 5𝑖 + 𝑗 + 11𝑘

SOLUTION: The GENERAL FORMULA for the CROSS PRODUCT can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The CROSS PRODUCT is one of two OPERATIONS that can be deployed using VECTOR MULTIPLICATION, the other operation being the DOT PRODUCT. In it’s most fundamental form, the CROSS PRODUCT takes two vectors, runs them through a standard process, and creates a third vector that is ORTHOGONAL (or PERPENDICULAR) to each of the original vectors.



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In this problem, our two vectors are given as: 𝐴 =< 2, 1, −1 > 𝐵 =< −3, 4, 1 > There are two routes we can take in solving this problem, the first will be using the GENERAL FORMULA that we are explicitly given in the NCEES Reference Handbook, which is written as: 𝐴 × 𝐵 =< 𝑎! 𝑏! − 𝑎! 𝑏! , 𝑎! 𝑏! − 𝑎! 𝑏! , 𝑎! 𝑏! − 𝑎! 𝑏! >= 𝐵 𝐴 𝑠𝑖𝑛𝜃 We have all the ELEMENTS defined for each of our VECTORS, but we don’t have an ANGLE between them. This is not a problem, the first half of this GENERAL FORMULA is enough to define our resulting VECTOR. Plugging in the ELEMENTS where outlined, we get: 𝐴 × 𝐵 =< 1(1) − (−1)(4), (−1)(−3) − 2(1), 2(4) − 1(−3) > Which results in a new VECTOR: 𝐴 × 𝐵 =< 5,1, 11 > Or in VECTOR NOTATION: 𝐴 × 𝐵 = 5𝑖 + 𝑗 + 11𝑘



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So we could stop right here, but let’s move forward with solving this problem with a second method, that being using a MATRIX and the METHOD OF COFACTORS. The FORMULA for the CROSS PRODUCT in MATRIX FORM can be referenced under the topic of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Illustrating the CROSS PRODUCT in a 3𝑥3 matrix, we get: 𝑖 𝐴 × 𝐵 = 𝑎! 𝑏!

𝑗 𝑎! 𝑏!

𝑘 𝑎! 𝑏!

Let’s get this set up with the information we are provided in our problem statement. Again, we are given: 𝐴 =< 2, 1, −1 > 𝐵 =< −3, 4, 1 > The first line of the determinant always contains the unit vectors in the direction of each of the Cartesian axes. Therefore: 𝑖

𝑗

𝑘

𝐴×𝐵 =

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The second line is always the scalar coefficients of each of the unit vectors for the first vector listed in the cross product. 𝑖 𝐴×𝐵 = 2

𝑗 1

𝑘 −1

The third line is always the scalar coefficients of each of the unit vectors for the second vector listed in the cross product. 𝑖 𝐴×𝐵 = 2 −3

𝑗 1 4

𝑘 −1 1

With our MATRIX now set up, we can deploy our knowledge of the METHOD OF COFACTORS to determine the CROSS PRODUCT. This method can generally be illustrated as: 𝑎! 𝐴×𝐵 = 𝑏 !

𝑎! 𝑎! 𝑖 − 𝑏! 𝑏!

𝑎! 𝑎! 𝑗 + 𝑏! 𝑏!

𝑎! 𝑏! 𝑘

Where: 𝑎 𝑐



𝑏 = 𝑎𝑑 − 𝑏𝑐 𝑑

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Carrying out the METHOD OF COFACTORS for our current scenario, we get:

𝐴×𝐵 =

1 4

−1 2 𝑖− 1 −3

−1 2 𝑗+ 1 −3

1 𝑘 4

Which concludes that the CROSS PRODUCT of the two vectors is: 𝐴 × 𝐵 = 5𝑖 + 𝑗 + 11𝑘 A couple quick notes: • It is important to note that in the METHOD OF COFACTORS, the TERMS ALTERNATE IN SIGN and each term is simply a 2𝑥2 matrix multiplied by a SCALAR. In our set up, observe how the first term will be put with a positive sign, our second term will be put with a negative sign, and our third term will be set against a positive sign…the sign is alternating…this is the point. • HINT: You can use your CALCULATOR to hack this problem in a fraction of the time. Check out the CALCULATOR WORKSHOPS page within PREPINEER to study just how we suggest you move forward with these types of problems. The correct answer choice is D. 𝟓𝒊 + 𝒋 + 𝟏𝟏𝒌



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