By E. Wolstenholme, W. J. Langford and E. A. Maxwell (Auth.)

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If θχ is the angle between a and m. SCALAR TRIPLE PRODUCT a . (b X 41 c) = a . am = a(a. m) a{a cos ^i) = where h ( = Ä C O S Ö I ) is the altitude and V the volume of parallelepiped OBDC AB' D' C, having OBCD as base. (c X a) = c . (a b) = K. X Hence a . (b χ c) = b . (c X a) = c . a and a . (b X c) = b . (c (I) a) X from (I) above, therefore Similarly and Further, since b . (c c . (a a. (b X a) = (b b) = (c X c ) . a. a). b. X c) = (a X b). c. X b X X c = -(c X (Π) J b), therefore a. (b X c) = —a.

E. U -1 12 -1 Examples Ua Find the scalar products of the following pairs of vectors: 813. 4i3. 3i3. 5i3. pairs of vectors: 5. - 2 i i - 12 - 3i3 and 4ii + 7i2 + 213. 6. 2ii - 312 + 5i3 and ii - 2i2 - 3i3. 7. ii + 3i2 - 813 and - 3 i i - 5i2 + 4i3. 8. 4ii + 1 2 - 1 3 and 5ii + 2i2 - -7 14/V38 21/V38 35/^/38 G = 7(16ii - 9i2 - i s W ^ lb wt. ft units 1. - 4 i i + 5i2 + 313 and 2ii + 7i2 2. 3ii + 9i2 - 2i3 and ii - 12 3. 5ii + 1 2 + 2ΐ3 and - 2 i i + 12 + 4. 2ii - 3i2 + 613 and 2ii - 3i2 Find the vector products of the following 13 7ΐ3.

Q + c . q + . . + a . r + ... etc. 8 By definition, if the vector b makes an angle Θ with the vector a, and η is a real number n(a . b) = nab cos θ also (n&). b = nab cos Θ and a . (nh) = nab cos Θ, Hence n(a . b) = («a). b = a . 9· Distributiye Law—Vector Product (i) Suppose the vectors a and b are represented by OA and 0 5 and suppose Π is the plane through Ο perpendicular to OA. Let BB' be the perpendicular from 5 to Π and let b' be the vector represented by 0B\ Then if b makes an angle θ mth a, b'=^b sin θ Fig.