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8.6.1 – The Dot Product (Inner Product)

8.6.1 – The Dot Product (Inner Product). So far, we have covered basic operations of vectors Addition/Subtraction Multiplication of scalars Writing vectors in various forms We will now talk about the last crucial operation. Dot Product. The product of two vectors will create a scalar

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8.6.1 – The Dot Product (Inner Product)

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  1. 8.6.1 – The Dot Product (Inner Product)

  2. So far, we have covered basic operations of vectors • Addition/Subtraction • Multiplication of scalars • Writing vectors in various forms • We will now talk about the last crucial operation

  3. Dot Product • The product of two vectors will create a scalar • The dot product of two vectors is given if u = {u1, u2} and v = {v1, v2} • The dot product may be positive, negative, or zero (similar to multiplication of real numbers)

  4. Example. Find the dot product if u = {-5, 2} and v = {3, -1} • Find each corresponding part

  5. Example. Find the dot product if u = {-5, 2} and v = {-5, 2}

  6. Example. Find the dot product if u = {-5,2} and v = {2, 5}

  7. Properties • With the dot product, we can derive certain properties • 1) u . v = v . u (commutative) • 2) 0 . u = 0 • 3) u . (v + w) = u . v + u . w (distribution) • 4) a(u . v) = (au) . v = u . (av) • 5) u . u = ||u||2

  8. Example. Find the quantity 3v . u if u = {-2, 3} and v = {4, 4}

  9. Example. Find the magnitude of the vector v if the dot product with itself is 12.

  10. Example. u . u = 80. Find ||u||.

  11. Dot Product Theorem • Similar to component form, we can talk about the dot product of vectors in terms of an angle • Let u and v be nonzero vectors, and ϴ be the smaller of the two angles formed by u and v; then,

  12. Example. Find the angle between the two vectors u = {5,4} and v = {3, 2}

  13. Example. Find the angle between the two vectors u = 5i + 2j, v = 4i + j

  14. Assignment • Pg. 678 • 1-23 odd

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