A problem to solve in Physics

Physics
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Consider the two vectors A~ = 3ˆi − 2 ˆj and B~ = −ˆi − 4 ˆj. Calculate (a) A~ + B~ , (b) A~ − B~ , (c) |A~ + B~ |, (a) |A~ − B~ |, and (e) the direction of A~ + B~ and A~ − B~ .

Oct 3rd, 2015

A = 3i - 2j

B = - i - 4j

Now

(a) A + B = (3i - 2j) + (- i - 4j)

                = 3i - 2j - i - 4j

                = 3i - i - 2j - 4j

                = 2i - 6j

(b) A - B = (3i - 2j) - (- i - 4j)

               = 3i - 2j + i + 4j

               = 3i + i - 2j + 4j

               = 4i + 2j

(c) |A + B| = |2i - 6j|

                  = √{(2)^2 + (-6)^2}

                  = √(4 + 36)

                  = √40

                  = √4 * 10

                  = √4 * √10

                  = 2√10

(d) |A - B| = |4i + 2j|

                 = √{(4)^2 + (2)^2}

                 = √(16 + 4)

                 = √20

                 = √4 * 5

                 = √4 * √5

                 = 2√5

(e) Direction of A + B

 we use the formula

Θ = tan^-1(y/x)

Now from A + B = 2i - 6j

x = 2    y = - 6

putting the values, we get

Θ = tan^-1 (- 6/2)

Θ = tan^-1 (- 3)

Θ = - 71.57 degrees

which is the direction of A + B

Now

Direction of A - B

using the formula

Θ = tan^-1(y/x)

From A - B = 4i + 2j

x = 4    y = 2

putting it in the formula, we get

Θ = tan^-1(2/4)

Θ = tan^-1(1/2)

Θ = tan^-1(0.5)

Θ = 26.57 degrees

which is the direction of A - B

Sep 17th, 2015

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