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Thread: Ned to solved some vector algebra proble.

  1. #1

    Ned to solved some vector algebra proble.

    Show That (a ⃗-b ⃗)×(a ⃗+b ⃗)=2a ⃗×b ⃗
    Show That a ⃗ ×(b ⃗+c ⃗)+b ⃗ ×(c ⃗+a ⃗)+c ⃗ ×(a ⃗+b ⃗)=0


    Find the scalar m such that the scalar product of i+j+k with unit vector parallel to the sum of 2i+4j-5k and mi+2j+3k is equal to unity.

    show that a=3i-2j+k, b=i-3j+5k, c=2i+j-4k form a right angled triangle
    Last edited by mathseeker; 02-18-2017 at 04:35 AM.

  2. #2
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    Well, first off, it looks like whatever you tried to copy and paste didn't come through as intended. In the first two problems, I see a bunch of white boxes throughout. Please reply with a corrected version of these problem statements. Additionally, please also comply with the rules as laid out in the Read Before Posting thread that's stickied at the top of each sub-forum (you did read it, right? ) and share with us any and all work you've done on this problems, even the parts you know for sure are wrong. Thank you.

  3. #3

    first two problems are

    Show That

    (vector a- vector b) X (vector a +vector b) = 2 vector a X vector b

    and

    Show That

    Vector a X (Vector b + Vector C) + Vector b X (Vector c + Vector a) + Vector C X (Vector a + Vector b) = 0

  4. #4
    Full Member MarkFL's Avatar
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    Suppose:

    [tex]\textbf{a}=a_1\textbf{i}+a_2\textbf{j}+ a_3\textbf{k}[/tex]

    [tex]\textbf{b}=b_1\textbf{i}+b_2\textbf{j}+ b_3\textbf{k}[/tex]

    Then, we have:

    [tex]\textbf{a}\times\textbf{b}= \left(a_2b_3-a_3b_2\right)\textbf{i}+ \left(a_3b_1-a_1b_3\right)\textbf{j}+ \left(a_1b_2-a_2b_1\right)\textbf{k}[/tex]

    Now, we are asked to verify the identity:

    [tex](\textbf{a}-\textbf{b}) \times(\textbf{a}+\textbf{b})= 2(\textbf{a}\times\textbf{b})[/tex]

    Can you compute the vector difference and sum needed on the left side?
    Living in the pools, They soon forget about the sea...— Rush, "Natural Science" (1980)

  5. #5
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    For the second problem, "Show that [tex]\vec{a}\times (\vec{b}+ \vec{c})+ \vec{b}\times(\vec{c}+ \vec{a})+ \vec{c}\times (\vec{b}+ \vec{a})= \vec{0}[/tex]", you could actually work out each term using [tex]\vec{a}= a_1\vec{i}+ a_2\vec{j}+ a_3\vec{k}[/tex], [tex]\vec{b}= b_1\vec{i}+ b_2\vec{j}+ b_3\vec{k}[/tex], [tex]\vec{a}= c_1\vec{i}+ c_2\vec{j}+ c_3\vec{k}[/tex] and [tex]\vec{c}= c_1\vec{i}+ c_2\vec{j}+ c_3\vec{k}[/tex] and the formula for the cross product MarkFL gave. But simpler would be to use the fact that "the cross product distributes over addition", [tex]\vec{a}\times (\vec{b}+ \vec{c})= \vec{a}\times\vec{b}+ \vec{a}\times \vec{c}[/tex], and the fact that the cross product is "anti-commutative", [tex]\vec{a}\times\vec{b}= -\vec{b}\times\vec{a}[/tex]. If you have not already learned those you can prove them, by using the formula MarkFL gave, once then use them repeatedly.

    The third and fourth problems are easier in that they do not require the cross product. Can we presume at leas that you do know what a "scalar product" is and how to show that two vectors are parallel? Or that you know how to find the length of a vector and that a triangle is a right triangle if and only if the lengths of its sides satisfy the "Pythagorean Theorem"?
    Last edited by HallsofIvy; 02-18-2017 at 07:29 AM.

  6. #6
    Quote Originally Posted by HallsofIvy View Post

    The third and fourth problems are easier in that they do not require the cross product. Can we presume at leas that you do know what a "scalar product" is and how to show that two vectors are parallel? Or that you know how to find the length of a vector and that a triangle is a right triangle if and only if the lengths of its sides satisfy the "Pythagorean Theorem"?
    BUT HERE SAYS THAT is equal to unity. WHAT'S THAT MEANS?

  7. #7

    Find the scalar m such that the scalar product of i+j+k with unit vector parallel to

    Find the scalar m such that the scalar product of i+j+k with unit vector parallel to the sum of 2i+4j-5k and mi+2j+3k is equal to unity.

  8. #8
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    Quote Originally Posted by mathseeker View Post
    BUT HERE SAYS THAT is equal to unity. WHAT'S THAT MEANS?
    That means that the scalar product = 1
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

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