{"id":3481,"date":"2023-10-05T10:40:55","date_gmt":"2023-10-05T10:40:55","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=3481"},"modified":"2023-10-05T10:42:53","modified_gmt":"2023-10-05T10:42:53","slug":"week-4-ss2-third-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-4-ss2-third-term-further-mathematics-notes\/","title":{"rendered":"Week 4 – SS2 Third Term Further Mathematics Notes"},"content":{"rendered":"

WEEK FOUR
\n<\/strong>TOPIC:MECHANICS (VECTOR GEOMETRY)
\n<\/strong>SCALAR OR DOT PRODUCT OF TWO VECTORS
\n<\/strong>The scalar or dot product of two vectors a and b is written as a.b and pronounced as (a dot b). Therefore, a.b =|a| |b| cos dot is defined as a.b = \u00a0\u00a0\u00a0\u00a0a b cos where is the angle between vectors a and b
\nIf a = a1 <\/sub>I + a2<\/sub>j and b = b1 <\/sub>I b2j
\nThus a .b = (a)1bi ii + ab2j I 1 + 2 bi I h +a2 b2 j
\nRecall that I and j are mutually perpendicular unit vector hence
\ni.i = |x| cos 0 =1
\ni.j = |x| cos 90 =0
\nj.i = |x| cos 90 =0
\nj.i =|x| cos 0 =1
\nHence, a.b =a1<\/sub>b1 <\/sub>+ a2 <\/sub>b2
\n<\/sub>Examples
\n<\/strong><\/p>\n

    \n
  1. \n
    Find the scale product of the following vectors 9i -2j + k and I \u2013 3j -4k\n<\/div>\n

    Solution:
    \n<\/strong>\u00a0\u00a0\u00a0\u00a0A=(9i- 2j +k) and b= (i-3j -4k)
    \n\u00a0\u00a0\u00a0\u00a0a.b = (9i-2j +k) (i-3j-4k)
    \n\u00a0\u00a0\u00a0\u00a0=9 (1) -2(-3) + 1(-4)=9+6-4a.b =11
    \n2.\u00a0\u00a0\u00a0\u00a0Let a = 3i+2j, b = -4i+2j and c = i+4j, calculate a.b, a.c and a. (b+c)
    \nSolution:
    \n<\/strong>\u00a0\u00a0\u00a0\u00a0I a.b = (3i + 2j ) (-4i+2j) = 3 (-4) +2(2)
    \n\u00a0\u00a0\u00a0\u00a0= -12+4
    \n\u00a0\u00a0\u00a0\u00a0=-8
    \nII a.c = (3i+2j) (I +4j)
    \n\u00a0\u00a0\u00a0\u00a0= 3 (1) + 2 (4)
    \n\u00a0\u00a0\u00a0\u00a0= 3+8\u00a0\u00a0\u00a0\u00a0= 11
    \nIII a.(b+c)
    \nFind (b+c) = -4i + 2j +i +4j
    \n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0=-3i +6j\n<\/li>\n

  2. \n
    (b+c) = (3i+2j) (-3i +6j)\n<\/div>\n

    \u00a0\u00a0\u00a0\u00a0=3(-3) + 2(6)
    \n\u00a0\u00a0\u00a0\u00a0= -9+12 = 3.<\/p>\n

    \u00a0PERPENDICULARITY OF VECTORS:
    \n<\/strong>If two vectors P and q are in perpendicular directions, thus p.q =0
    \n Example 1: show that the vectors p = 3i+ 2j and q= -2i + 3j are perpendicular.
    \nSolution:
    \nP:q = (3i+2j) (-2i +3j)
    \n=3(-2) + 2(3)
    \n=-6+ =0
    \nSince p.q=0, then the vectors p and q are perpendicular.
    \n2. If p= 4i + kj and q=2i \u2013 3j are perpendicular, find the value of k, where k is a scalar..
    \nSolution:
    \n<\/strong>p.q=0
    \n(4i+kj)(2i-3j)=0
    \n4(2) + k(-3)=0
    \n-3k=-8
    \nK=8\/3.<\/p>\n

    \u00a0EQUAL VECTORS:<\/strong> Vectors p ad q are equal if p is equal to q.
    \nExample: find the value of the scalar K for which the vectors 2ki + 3j and 8i+kj
    \nSolution:
    \n2ki +3j = 8i +kj
    \nHence, 2ki =8i,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a03j=
    \n2k = 8\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a012=3k
    \nK=8\/2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0k=12\/3
    \nK=4\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0k=4
    \nEVALUATION
    \n<\/strong><\/li>\n

  3. The vectors AB and C are -2i+6j-3k and -2i-3j+6k respectively. Find the scalar product AB.AC\n<\/li>\n
  4. \n
    Find the value of the scalar A for which the pairs of vectors 5i +3j and 2i-4Aj are perpendicular.\n<\/div>\n

    \u00a0ANGLES BETWEEN TWO VECTORS
    \n<\/strong> Is the angle between two vectors and from dot product where a.b=|a|b| cos . Hence, Cos
    \nWhere = Magnitude of vector a= 2<\/sup>1 <\/sub>+2<\/sup>2<\/sub>
    \n\t\t\t\t\u00a0\u00a0\u00a0\u00a0 |b| =Magnitude of vector b=2<\/sup>1 <\/sub>+2<\/sup>2<\/sub><\/p>\n

    \u00a0Example:
    \nFind the angle between the vectors pp=2i \u2013 2j + k and q=12i +4j \u2013 3k
    \nSolution:
    \n<\/strong>Cos =
    \np.q= (2i-2j+k)(12i+4j-3k)
    \n\u00a0\u00a0\u00a0\u00a0=2(12) -2(4)_1(3)
    \n\u00a0\u00a0\u00a0\u00a0= 24-8-3
    \n\u00a0\u00a0\u00a0\u00a0=13
    \n|p|=2<\/sup> + (-2)2<\/sup> + 12<\/sup>=+4+1 ==3
    \n|q|= 2<\/sup> + 42<\/sup> + (-3) = 144+16+9= 169 =13
    \nCos
    \nCos =1\/3. =Cos-1<\/sup> (1\/3)<\/p>\n

    \u00a0DIRECTION COSINES A VECTOR:
    \n<\/strong>The direction is specified by the angles which the vector makes wit x and y axes. If we represent these angles by and respectively then,
    \nCos = \u00a0\u00a0\u00a0\u00a0Cos =
    \nExample: find the direction cosine of the vector 4i + 3J \u2013 11k
    \nSolution:
    \n<\/em><\/strong>Let a = 4i + 3J-11k
    \n\u00a0\u00a0\u00a0\u00a0|a|= 2<\/sup>\u00a0\u00a0\u00a0\u00a0+ y2<\/sup> +z2<\/sup> = |=2<\/sup> + 2<\/sup> + (-11)2<\/sup> =
    \nDirection cosine, Cos Cos =Cos= <\/p>\n

    \u00a0EVALUATION
    \n<\/strong><\/li>\n

  5. Find the angle between the vectors 2i + 3j +6k and 3i+4j+12k\n<\/li>\n
  6. \n
    Find the direction cosine of vector a = 10i- j+2k\n<\/div>\n

    \u00a0EVALUATION<\/strong>: find the projection of the vector a on the vector b if a=5i-4j+2k and b=6i \u2013 j +3k <\/p>\n

    \u00a0GENERAL REVISION EVALUATION
    \n<\/strong><\/li>\n

  7. Given that a=4i \u2013 2j +k, b=2i \u2013 j +3k and c=5i +2k find (i) (a+b)c\u00a0\u00a0\u00a0\u00a0 (ii) a=c+b.c\n<\/li>\n
  8. \n
    If a = 4i \u2013 2j +k, b=6i +5j find (i) the unit vector in direction of b. (ii) the projection of a on b (iii) the unit ve4xtor in the direction of a (iv) the projection of b on a.\n<\/div>\n

    \u00a0READING ASSIGNMENT:<\/strong> Read vector Geometry, Further Mathematics project II page 236-240 <\/p>\n

    \u00a0WEEKEND ASSIGNMENT
    \n<\/strong><\/li>\n

  9. If = 3i + 4j and b=gi +2k are perpendicular, what is the value of g? A.-4 B.3 C.-8\/3\n\t\t\t\t<\/li>\n
  10. Find the value of the scalar k for which the vectors ki + 8j and 3i + are equal. \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0A. 3\u00a0\u00a0\u00a0\u00a0B.6\u00a0\u00a0\u00a0\u00a0C.9\n\t\t\t\t<\/li>\n
  11. Find the projection of the vector a on the vector if a = 4i + 6j and b=3i-2j. \u00a0\u00a0\u00a0\u00a0A.-3\\ 52 B.5\\ 13 C. 0\n\t\t\t\t<\/li>\n
  12. Calculate the angle between a = -4i +2j and b =I -3j. A.450<\/sup> B.600<\/sup> C.1350<\/sup>\n\t\t\t\t<\/li>\n
  13. \n
    Find the scalar product of vectors \u2013 2i-3j and 4i +5j? A.-23 B.23 C.7\n\t\t\t\t\t<\/div>\n

    \u00a0THEORY
    \n<\/strong>1a) Given that a = 4i \u2013 5j + 2k and b = -7i + 3j \u2013 6k find the scalar product of a and b (b) find the direction cosine 2a + 3b
    \n2 ) Find the angle between p = 6i + 2j \u2013 4k and q = 9i + 5j <\/p>\n

    \u00a0
    \n\t\t\t\t\t<\/strong>\u00a0<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    WEEK FOUR TOPIC:MECHANICS (VECTOR GEOMETRY) SCALAR OR DOT PRODUCT OF TWO VECTORS The scalar or…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,275],"tags":[],"class_list":["post-3481","post","type-post","status-publish","format-standard","hentry","category-posts","category-third-term-ss2-further-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/3481","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/comments?post=3481"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/3481\/revisions"}],"predecessor-version":[{"id":3482,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/3481\/revisions\/3482"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=3481"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=3481"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=3481"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}