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Vector Refresher Part 2 • Vector Addition • Multiplication by Scalar Quantities • Graphical Representations • Analytical Methods Vector Addition • A lot of times, it’s important to know the resultant of a set of vectors • The resultant vector can be described as the sum of all the vector quantities you wish to evaluate • We’ll look at graphical and analytical techniques for doing this Graphical Methods For Vector Addition • 2 ways to look at vector addition graphically • Parallelogram method A B Connect the tails of both vectors to start. Graphical Methods For Vector Addition • 2 ways to look at vector addition graphically • Parallelogram method A B Create a parallelogram with each vector as one of the sides. Graphical Methods For Vector Addition • 2 ways to look at vector addition graphically • Parallelogram method A B The resultant goes from the place where 2 tails meet to the place where 2 heads meet A+B Graphical Methods For Vector Addition • 2 ways to look at vector addition graphically • Head-to-tail method A B With this method, the vectors are added by placing the tail of a vector at the head of another one to make a continuous path. Graphical Methods For Vector Addition • 2 ways to look at vector addition graphically • Head-to-tail method A B The addition of these vectors results in a vector that starts at the tail of the first vector and ends at the head of the final vector A+B Graphical Methods For Vector Addition • 2 ways to look at vector addition graphically • Head-to-tail method B A C If we had a 3rd vector, we could add it to the previous result A+B Graphical Methods For Vector Addition • 2 ways to look at vector addition graphically • Head-to-tail method B A C If we had a 3rd vector, we could add it to the previous result A+B A+ B+C Multiplying a Vector By a Scalar When a vector is multiplied by a scalar, the length of the vector can be elongated or shortened. If the scalar is a negative number, the direction is reversed. Multiplying a Vector By a Scalar When a vector is multiplied by a scalar, the length of the vector can be elongated or shortened. If the scalar is a negative number, the direction is reversed. A 2A Multiplying a vector by 2 yields a vector with the SAME DIRECTION and twice the length Multiplying a Vector By a Scalar When a vector is multiplied by a scalar, the length of the vector can be elongated or shortened. If the scalar is a negative number, the direction is reversed. A 1 A 2 Multiplying a vector by 1/2 yields a vector with the SAME DIRECTION and half the length Multiplying a Vector By a Scalar When a vector is multiplied by a scalar, the length of the vector can be elongated or shortened. If the scalar is a negative number, the direction is reversed. A -A Multiplying a vector by -1 yields a vector with the OPPOSITE DIRECTION and the same length Example 1 Draw A - 3B 2 Start by multiplying vector A by 1/2 A B Example 1 Draw A - 3B 2 Start by multiplying vector A by 1/2 1 A 2 A B Example 1 Draw A - 3B 2 Next, multiply vector B by -3 - 3B 1 A 2 A B Example 1 Draw A - 3B 2 Now add these vectors together - 3B 1 A 2 A B Example 1 Draw A - 3B 2 A The resultant vector is shown in green - 3B 1 A 2 1 A - 3B 2 B Analytical Method for Vector Addition When vectors are added to each other, the components in each direction are summed. U = aiˆ + bjˆ + ckˆ V = diˆ + ejˆ + fkˆ U +V = (a + d)iˆ + (b + e) jˆ + (c + f )kˆ Analytical Method for Vector Addition When vectors are added to each other, the components in each direction are summed. U = aiˆ + bjˆ + ckˆ V = diˆ + ejˆ + fkˆ U +V = (a + d)iˆ + (b + e) jˆ + (c + f )kˆ Analytical Method for Vector Addition When vectors are added to each other, the components in each direction are summed. U = aiˆ + bjˆ + ckˆ V = diˆ + ejˆ + fkˆ U +V = (a + d)iˆ + (b + e) jˆ + (c + f )kˆ Multiplying a Vector By a Scalar When multiplying a vector by a scalar, the scalar must be applied to each component of the vector Thus, if we multiply V we get: = aiˆ + bjˆ + ckˆ by a scalar “k”, kV = kaiˆ + kbjˆ + kckˆ Example Determine the resultant force caused by the following 3 force vectors. F1 = éë16iˆ +10 jˆ + 2kˆùûlbs F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs 1 F3 = 2F1 - F2 3 The first thing to do is determine F3 ( 2F1 = 2 éë16iˆ +10 jˆ + 2kˆùûlbs ) Example Determine the resultant force caused by the following 3 force vectors. F1 = éë16iˆ +10 jˆ + 2kˆùûlbs F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs 1 F3 = 2F1 - F2 3 The first thing to do is determine F3 ( 2F1 = 2 éë16iˆ +10 jˆ + 2kˆùûlbs ) 2F1 = éë(2·16)iˆ + (2·10) jˆ + (2·2)kˆùûlbs Example Determine the resultant force caused by the following 3 force vectors. F1 = éë16iˆ +10 jˆ + 2kˆùûlbs F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs 1 F3 = 2F1 - F2 3 The first thing to do is determine F3 ( 2F1 = 2 éë16iˆ +10 jˆ + 2kˆùûlbs ) 2F1 = éë(2·16)iˆ + (2·10) jˆ + (2·2)kˆùûlbs 2F1 = éë32iˆ + 20 jˆ + 4kˆùûlbs Example Determine the resultant force caused by the following 3 force vectors. The first thing to do is determine F3 F1 = éë16iˆ +10 jˆ + 2kˆùûlbs 2F1 = éë32iˆ + 20 jˆ + 4kˆùûlbs F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs 1 F3 = 2F1 - F2 3 1 1 é ˆ F2 = ë6i + 6 jˆ + 9kˆùûlbs 3 3 ( ) Example Determine the resultant force caused by the following 3 force vectors. The first thing to do is determine F3 F1 = éë16iˆ +10 jˆ + 2kˆùûlbs 2F1 = éë32iˆ + 20 jˆ + 4kˆùûlbs F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs 1 F3 = 2F1 - F2 3 1 1 é ˆ F2 = ë6i + 6 jˆ + 9kˆùûlbs 3 3 éæ 1 ö ˆ æ 1 ö ˆ æ 1 ö ˆù 1 F2 = êç ·6 ÷ i + ç ·6 ÷ j + ç ·9 ÷ k úlbs 3 ëè 3 ø è 3 ø è 3 ø û ( ) Example Determine the resultant force caused by the following 3 force vectors. The first thing to do is determine F3 F1 = éë16iˆ +10 jˆ + 2kˆùûlbs 2F1 = éë32iˆ + 20 jˆ + 4kˆùûlbs F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs 1 F3 = 2F1 - F2 3 1 1 é ˆ F2 = ë6i + 6 jˆ + 9kˆùûlbs 3 3 éæ 1 ö ˆ æ 1 ö ˆ æ 1 ö ˆù 1 F2 = êç ·6 ÷ i + ç ·6 ÷ j + ç ·9 ÷ k úlbs 3 ëè 3 ø è 3 ø è 3 ø û 1 F2 = éë2iˆ + 2 jˆ + 3kˆùûlbs 3 ( ) Example Determine the resultant force caused by the following 3 force vectors. F1 = éë16iˆ +10 jˆ + 2kˆùûlbs F2 = éë6iˆ + 6 jˆ + 9 kˆùûlbs 1 F3 = 2F1 - F2 3 The first thing to do is determine F3 2F1 = éë32iˆ + 20 jˆ + 4kˆùûlbs 1 F2 = éë2iˆ + 2 jˆ + 3kˆùûlbs 3 1 2F1 - F2 = éë(32 - 2)iˆ + (20 - 2) jˆ + (4 - 3)kˆùûlbs 3 1 2F1 - F2 = éë30iˆ +18 jˆ + kˆùûlbs 3 F3 = éë30iˆ +18 jˆ + kˆùûlbs Example Determine the resultant force caused by the following 3 force vectors. Now we can add the 3 vectors together to determine the resultant force F1 = éë16iˆ +10 jˆ + 2kˆùûlbs F2 = éë6iˆ + 6 jˆ + 9kˆùûlbs F3 = éë30iˆ +18 jˆ + kˆùûlbs FR = F1 + F2 + F3 = (16 + 6 + 30)iˆ Example Determine the resultant force caused by the following 3 force vectors. Now we can add the 3 vectors together to determine the resultant force F1 = éë16iˆ +10 jˆ + 2kˆùûlbs F2 = éë6iˆ + 6 jˆ + 9kˆùûlbs F3 = éë30iˆ +18 jˆ + kˆùûlbs FR = F1 + F2 + F3 = (16 + 6 + 30)iˆ + (10 + 6 +18) jˆ Example Determine the resultant force caused by the following 3 force vectors. Now we can add the 3 vectors together to determine the resultant force F1 = éë16iˆ +10 jˆ + 2kˆùûlbs F2 = éë6iˆ + 6 jˆ + 9kˆùûlbs F3 = éë30iˆ +18 jˆ + kˆùûlbs FR = F1 + F2 + F3 = (16 + 6 + 30)iˆ + (10 + 6 +18) jˆ + (2 + 9 +1)kˆ Example Determine the resultant force caused by the following 3 force vectors. Don’t forget the units, they’re as important to the answer as the numbers F1 = éë16iˆ +10 jˆ + 2kˆùûlbs are F2 = éë6iˆ + 6 jˆ + 9kˆùûlbs F3 = éë30iˆ +18 jˆ + kˆùûlbs FR = F1 + F2 + F3 = éë(16 + 6 + 30)iˆ + (10 + 6 +18) jˆ + (2 + 9 +1)kˆùûlbs Example Determine the resultant force caused by the following 3 force vectors. Don’t forget the units, they’re as important to the answer as the numbers F1 = éë16iˆ +10 jˆ + 2kˆùûlbs are F2 = éë6iˆ + 6 jˆ + 9kˆùûlbs F3 = éë30iˆ +18 jˆ + kˆùûlbs FR = F1 + F2 + F3 = éë(16 + 6 + 30)iˆ + (10 + 6 +18) jˆ + (2 + 9 +1)kˆùûlbs FR = éë52iˆ + 34 jˆ +12kˆùûlbs