Product Development Essentials
In the process of product development, key aspects like understanding the market, defining the product, creating a bill of materials, and designing the product play a crucial role. This comprehensive guide covers the foundational steps from ideation to production, emphasizing the importance of strategic decision-making and planning to ensure a successful product launch.
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28 February 2025 Vector product of two vectors LO: To understand and perform the product vector or cross product. www.mathssupport.org
Vector (cross) product There are two ways to take the product of a pair of vectors. The dot product The cross product We have discussed the dot product in another lesson. The cross product is defined only for three-dimensional vectors. If u and v are two three-dimensional vectors, then their cross product, written as u v pronounced u cross v is another three-dimensional vector. www.mathssupport.org www.mathssupport.org
Vector (cross) product When we define this cross-product vector u v keep in mind that u v is a vector that is perpendicular to both u and v Since the cross product must be perpendicular to the two unit vectors, it must be equal to the other unit vector or the opposite of that unit vector. i j j k = i k i = j This little cycle diagram can help you remember these results. = k i k j www.mathssupport.org www.mathssupport.org
Vector (cross) product What about i k ? It must be equal to the opposite of the other unit vector. k i = j i k = j k j = i j i j k = i i j = k = k Finally, the cross product of any vector with itself is the zero vector a a = 0 . In particular, the cross product of any standard unit vector with itself is the zero vector. With this rules we are going to find an expression for the vector product. www.mathssupport.org www.mathssupport.org
Vector (cross) product Given u = u1i + u2j + u3k and v = v1i + v2j + v3k Find u v u v = (u1i + u2j + u3k) = (u1v1)(i i) a a = 0 i j = k i k = j j i = k (v1i + v2j + v3k) + (u1v3)(i k) + (u1v2)(i j) Simplifying Expanding brackets + (u2v1)(j i) + (u3v1)(k i) + (u2v2)(j j) + (u3v2)(k j) + (u3v3)(k k) + (u2v3)(j k) j k = i k i = j k j = i = (u1v2)k + (u3v1)j (u3v2)i = (u2v3 u3v2)i+ (u3v1 u1v3)j + (u1v2 u2v1)k u v = (u2v3 u3v2)i+ (u3v1 u1v3)j + (u1v2 u2v1)k (u1v3)j (u2v1)k + (u2v3)i Simplifying www.mathssupport.org www.mathssupport.org
Vector (cross) product Given u = 3i 3j +k and v = 4i + 6j + 2k Find u v u v = (u2v3 u3v2)i+ (u3v1 u1v3)j + (u1v2 u2v1)k = ((-3)(2) (1)(6))i+ ((1)(4) (3)(2))j+ ((3)(6) (-3)(4)) k = 12i 2j + 20k www.mathssupport.org www.mathssupport.org
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