How to Add Vectors

How to Add Vectors Do you remember the basic rule for adding vectors?

How to Add Vectors Tip to Tail Rule

How to Add Vectors Tip to Tail Rule Arrange the tip of the first vector with the tail of the second vector, then the tip of the second vector with the tail of the third vector and so on...

How to Add Vectors Tip to Tail Rule Arrange the tip of the first vector with the tail of the second vector, then the tip of the second vector with the tail of the third vector and so on... Draw a resultant vector from start to finish

How to Add Vectors Tip to Tail Rule Arrange the tip of the first vector with the tail of the second vector, then the tip of the second vector with the tail of the third vector and so on... Draw a resultant vector from start to finish Example: Add a + b + c c a b

How to Add Vectors Tip to Tail Rule a + b + c a c b

How to Add Vectors Tip to Tail Rule a + b + c a c b tail tip

How to Add Vectors Tip to Tail Rule a + b + c a c b tail tip

How to Add Vectors Tip to Tail Rule a + b + c a + b + c a c b tail tip

Does it matter in which order we add vectors? Does a+b+c=c+b+a or a+c+b? tail tip

Does it matter in which order we add vectors? Does a+b+c=c+b+a or a+c+b? tail tip

Does it matter in which order we add vectors? Does a+b+c=c+b+a or a+c+b? tail tip c+b+a

Does it matter in which order we add vectors? Does a+b+c=c+b+a or a+c+b? tail tip c+b+a

Does it matter in which order we add vectors? Does a+b+c=c+b+a or a+c+b? tail tip a+c+b c+b+a

Vector Addition Coda

Vector Addition Coda For non-collinear vectors not along the same line, the tip-to-tail rule is the fundamental way to add vectors

Vector Addition Coda For non-collinear vectors not along the same line, the tip-to-tail rule is the fundamental way to add vectors For vectors along the same line, as with the BIG FIVE equations, vectors can be added or subtracted by representing them as integers

Vector Addition Coda For non-collinear vectors not along the same line, the tip-to-tail rule is the fundamental way to add vectors For vectors along the same line, as with the BIG FIVE equations, vectors can be added or subtracted by representing them as integers Order doesn't matter when adding vectors even with the tip-to-tail rule

Vector Addition Coda For non-collinear vectors not along the same line, the tip-to-tail rule is the fundamental way to add vectors For vectors along the same line, as with the BIG FIVE equations, vectors can be added or subtracted by representing them as integers Order doesn't matter when adding vectors even with the tip-to-tail rule Two vectors are equal if their length or magnitude is the same AND their direction in space is the same

Vector Addition Coda For non-collinear vectors not along the same line, the tip-to-tail rule is the fundamental way to add vectors For vectors along the same line, as with the BIG FIVE equations, vectors can be added or subtracted by representing them as integers Order doesn't matter when adding vectors even with the tip-to-tail rule Two vectors are equal if their length or magnitude is the same AND their direction in space is the same We can't add different vector quantities ... example: We can't add a velocity vector and a displacement vector. We can only add vectors if the type of quantity and the units used are the same.

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Why can't we say the answer is 3+4 =7.0 m [NW] ?

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Why can't we say the answer is 3+4 =7.0 m [NW] ? These quantities have magnitude and direction, and are therefore vector quantities. Adding vectors like displacement, acceleration or velocity has different rules than adding scalar quantities like mass and speed.

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Why can't we say the answer is 3+4 =7.0 m [NW] ? These quantities have magnitude and direction, and are therefore vector quantities. Adding vectors like displacement, acceleration or velocity has different rules than adding scalar quantities like mass and speed. Why can't we say -3 +4 = +1 or 1.0 m [N] ?

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Why can't we say the answer is 3+4 =7.0 m [NW] ? These quantities have magnitude and direction, and are therefore vector quantities. Adding vectors like displacement, acceleration or velocity has different rules than adding scalar quantities like mass and speed. Why can't we say -3 +4 = +1 or 1.0 m [N] ? We can represent vectors as integers, but only if the directions of each vector are along a line. [N] and [W] are not collinear.

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule 4.0 m [N] 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule 4.0 m [N] 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule Δd 4.0 m [N] 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule │Δd│ = ? Δd 4.0 m [N] 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule │Δd│ = ? = √ ( 32 + 42 ) Δd 4.0 m [N] 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule │Δd│ = ? = √ ( 32 + 42 ) = 5.0 m Δd 4.0 m [N] 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule │Δd│ = ? = √ ( 32 + 42 ) = 5.0 m Δd 4.0 m [N] Θ 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule │Δd│ = ? = √ ( 32 + 42 ) = 5.0 m Θ = ? Δd 4.0 m [N] Θ 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule │Δd│ = ? = √ ( 32 + 42 ) = 5.0 m Θ = ? Δd 4.0 m [N] Θ So/hCa/hTo/a 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule │Δd│ = ? = √ ( 32 + 42 ) = 5.0 m Θ = ? Θ = tan-1( o/a ) Δd 4.0 m [N] Θ So/hCa/hTo/a 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule │Δd│ = ? = √ ( 32 + 42 ) = 5.0 m Θ = ? Θ = tan-1( o/a ) Θ = tan-1( 4/3 ) Δd 4.0 m [N] Θ So/hCa/hTo/a 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule │Δd│ = ? = √ ( 32 + 42 ) = 5.0 m Θ = ? Θ = tan-1( o/a ) Θ = tan-1( 4/3 ) = 53° Δd 4.0 m [N] Θ So/hCa/hTo/a 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule │Δd│ = ? = √ ( 32 + 42 ) = 5.0 m Θ = ? Θ = tan-1( o/a ) Θ = tan-1( 4/3 ) = 53° Answer: Δd 4.0 m [N] Θ So/hCa/hTo/a 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule │Δd│ = ? = √ ( 32 + 42 ) = 5.0 m Θ = ? Θ = tan-1( o/a ) Θ = tan-1( 4/3 ) = 53° Answer: 5.0 m [W53°N] or [ ? ] Δd 4.0 m [N] Θ So/hCa/hTo/a 3.0 m [W]

Adding vectors 90° to each other Example #1: 3.0 m [W] + 4.0 m [N] Apply the tip-to-tail rule │Δd│ = ? = √ ( 32 + 42 ) = 5.0 m Θ = ? Θ = tan-1( o/a ) Θ = tan-1( 4/3 ) = 53° Answer: 5.0 m [W53°N] or [N37°W] Δd 4.0 m [N] Θ So/hCa/hTo/a 3.0 m [W]

Adding vectors 90° to each other Example #2: 20.8 m/s [S] + 12.0 m/s [E] Try this, yourself !

Adding vectors 90° to each other Example #2: 20.8 m/s [S] + 12.0 m/s [E] Try this, yourself ! v = √(20.82 + 122) = 24.0 m/s Θ = tan-1( 12 / 20.8 ) = 30.0° Answer: 24.0 m/s [S30.0°E] Or [E60.0°S] v 20.8 m/s [S] Θ 12.0 m/s [E]

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W]

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] 8.00 N [N53.1°W] 53.1° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] 8.00 N [N53.1°W] 53.1° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] 8.00 N [N53.1°W] F 53.1° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] │F │= ? 8.00 N [N53.1°W] F 53.1° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] │F │= ? SAS use cos law 8.00 N [N53.1°W] F 53.1° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] │F │= ? SAS use cos law 8.00 N [N53.1°W] F 53.1° β 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] │F │= ? SAS use cos law 8.00 N [N53.1°W] F 53.1° 90-53.1=36.9° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] │F │= ? SAS use cos law 8.00 N [N53.1°W] F 53.1° 90-53.1=36.9° α 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] │F │= ? SAS use cos law 8.00 N [N53.1°W] F 53.1° 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] │F │= ? SAS use cos law │F │2 = ? 8.00 N [N53.1°W] F 53.1° 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] │F │= ? SAS use cos law │F │2 =22 + 82 – (2)(8)(2)cos(66.9°) 8.00 N [N53.1°W] F 53.1° 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] │F │= ? SAS use cos law │F │2 =22 + 82 – (2)(8)(2)cos(66.9°) │F │= ? 8.00 N [N53.1°W] F 53.1° 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] │F │= ? SAS use cos law │F │2 =22 + 82 – (2)(8)(2)cos(66.9°) │F │= 7.45 N 8.00 N [N53.1°W] F 53.1° 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] │F │= ? SAS use cos law │F │2 =22 + 82 – (2)(8)(2)cos(66.9°) │F │= 7.45 N What else is needed to specify the vector ? 8.00 N [N53.1°W] F 53.1° 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] │F │= ? SAS use cos law │F │2 =22 + 82 – (2)(8)(2)cos(66.9°) │F │= 7.45 N What else is needed to specify the vector ? direction specified by an angle at the start of the resultant γ 8.00 N [N53.1°W] F 53.1° 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] │F │= ? SAS use cos law │F │2 =22 + 82 – (2)(8)(2)cos(66.9°) │F │= 7.45 N What else is needed to specify the vector ? direction specified by an angle at the start of the resultant γ How can we find γ ? 8.00 N [N53.1°W] F 53.1° 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the Sine Law: sin γ / 8 = ? 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the Sine Law: sin γ / 8 = sin 66.9/ 7.45 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the Sine Law: sin γ / 8 = sin 66.9/ 7.45 sin γ = ? 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the Sine Law: sin γ / 8 = sin 66.9/ 7.45 sin γ = 8 sin 66.9/ 7.45 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the Sine Law: sin γ / 8 = sin 66.9/ 7.45 sin γ = 8 sin 66.9/ 7.45 γ = sin-1 ( 8 sin 66.9/ 7.45 ) 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the Sine Law: sin γ / 8 = sin 66.9/ 7.45 sin γ = 8 sin 66.9/ 7.45 γ = sin-1 ( 8 sin 66.9/ 7.45 ) γ = 81.0° or ??? 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the Sine Law: sin γ / 8 = sin 66.9/ 7.45 sin γ = 8 sin 66.9/ 7.45 γ = sin-1 ( 8 sin 66.9/ 7.45 ) γ = 81.0° or 180 – 81 = 99.0° 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the Sine Law: sin γ / 8 = sin 66.9/ 7.45 sin γ = 8 sin 66.9/ 7.45 γ = sin-1 ( 8 sin 66.9/ 7.45 ) γ = 81.0° or 180 – 81 = 99.0° Which value of γ is correct? 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the Sine Law: sin γ / 8 = sin 66.9/ 7.45 sin γ = 8 sin 66.9/ 7.45 γ = sin-1 ( 8 sin 66.9/ 7.45 ) γ = 81.0° or 180 – 81 = 99.0° Which value of γ is correct? We don't know. Another method must be used here!! 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the Sine Law: sin γ / 8 = sin 66.9/ 7.45 sin γ = 8 sin 66.9/ 7.45 γ = sin-1 ( 8 sin 66.9/ 7.45 ) γ = 81.0° or 180 – 81 = 99.0° Note well: If the angle is opposite the longest side, the angle could be acute or obtuse. In this case, use the cos law. 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the Sine Law: sin γ / 8 = sin 66.9/ 7.45 sin γ = 8 sin 66.9/ 7.45 γ = sin-1 ( 8 sin 66.9/ 7.45 ) γ = 81.0° or 180 – 81 = 99.0° Note well: If the angle is opposite the longest side, the angle could be acute or obtuse. In this case, use the cos law. Also note: you could solve for δ using sin law, then get γ δ 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the cos Law to find γ: δ 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the cos Law to find γ: Focus on side opposite γ δ 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the cos Law to find γ: Focus on side opposite γ 82 = ?? δ 8.00 N [N53.1°W] F 53.1° 7.45 N 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the cos Law to find γ: Focus on side opposite γ 82 = 22 + 7.452 – 2(2)(7.45)cosγ δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the cos Law to find γ: Focus on side opposite γ 82 = 22 + 7.452 – 2(2)(7.45)cosγ 64= 4 + 55.5 -29.8cosγ δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the cos Law to find γ: Focus on side opposite γ 82 = 22 + 7.452 – 2(2)(7.45)cosγ 64= 4 + 55.5 -29.8cosγ +29.8cosγ= 4 + 55.5 - 64 δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the cos Law to find γ: Focus on side opposite γ 82 = 22 + 7.452 – 2(2)(7.45)cosγ 64= 4 + 55.5 -29.8cosγ +29.8cosγ= 4 + 55.5 – 64 29.8cosγ= -4.5 δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the cos Law to find γ: Focus on side opposite γ 82 = 22 + 7.452 – 2(2)(7.45)cosγ 64= 4 + 55.5 -29.8cosγ +29.8cosγ= 4 + 55.5 – 64 29.8cosγ= -4.5 γ = ??? δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the cos Law to find γ: Focus on side opposite γ 82 = 22 + 7.452 – 2(2)(7.45)cosγ 64= 4 + 55.5 -29.8cosγ +29.8cosγ= 4 + 55.5 – 64 29.8cosγ= -4.5 γ = cos-1(-4.5/29.8) δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 2.00 N [E30.0°N] γ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the cos Law to find γ: Focus on side opposite γ 82 = 22 + 7.452 – 2(2)(7.45)cosγ 64= 4 + 55.5 -29.8cosγ +29.8cosγ= 4 + 55.5 – 64 29.8cosγ= -4.5 γ = cos-1(-4.5/29.8) = 98.7° δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 98.7° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the cos Law to find γ: Focus on side opposite γ 82 = 22 + 7.452 – 2(2)(7.45)cosγ 64= 4 + 55.5 -29.8cosγ +29.8cosγ= 4 + 55.5 – 64 29.8cosγ= -4.5 γ = cos-1(-4.5/29.8) = 98.7° Why can't 98.7° be used to specify the direction of F ? δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 98.7° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] Let's try the cos Law to find γ: Focus on side opposite γ 82 = 22 + 7.452 – 2(2)(7.45)cosγ 64= 4 + 55.5 -29.8cosγ +29.8cosγ= 4 + 55.5 – 64 29.8cosγ= -4.5 γ = cos-1(-4.5/29.8) = 98.7° Note 98.7°cannot specify the direction of F b/c it is not with respect to any standard direction N,E,S,W δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 98.7° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] How can we write the final answer? δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 98.7° 2.00 N [E30.0°N] 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] How can we write the final answer? δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 98.7° 2.00 N [E30.0°N] Θ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] How can we write the final answer? Θ = ??? δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 98.7° 2.00 N [E30.0°N] Θ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] How can we write the final answer? Θ = 180 – 30.0 – 98.7 δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 98.7° 2.00 N [E30.0°N] Θ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] How can we write the final answer? Θ = 180 – 30.0 – 98.7 = 51.3° δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 98.7° 2.00 N [E30.0°N] Θ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] How can we write the final answer? Θ = 180 – 30.0 – 98.7 = 51.3° How do we write the final answer? F = ???? δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 98.7° 2.00 N [E30.0°N] Θ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] How can we write the final answer? Θ = 180 – 30.0 – 98.7 = 51.3° How do we write the final answer? F = 7.45 N [W51.3°N] or [ ??? ] δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 98.7° 2.00 N [E30.0°N] Θ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #1: 2.00 N [E30.0°N] + 8.00 N [N53.1°W] How can we write the final answer? Θ = 180 – 30.0 – 98.7 = 51.3° How do we write the final answer? F = 7.45 N [W51.3°N] or [ ??? ] F = 7.45 N [W51.3°N] or [N38.7W] δ 8.00 N [N53.1°W] 7.45 N 53.1° F 90-53.1=36.9° 30.0° 98.7° 2.00 N [E30.0°N] Θ 30.0°

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #2: 15.0 m/s [N30.0°E] + 55.0 m/s [E53.1°S] Try this yourself !!

Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method Example #2: 15.0 m/s [N30.0°E] + 55.0 m/s [E53.1°S] |V|2 = 152+552-2(15)(55)cos(66.9) |V| = 51.0 m/s Resultant is opposite longest side, so Ψ could be obtuse or acute. Use cos law 552 = 152+512-2(15)(51)cosΨ 2(15)(51)cosΨ = -552+ 152+512 Ψ = 97.5° Θ = 180°-(30.0°+Ψ )= 52.5° Answer: 51.0 m/s [S52.5°E] or [E37.5] 15.0 m/s 53.1° 30° 30° Ψ 55.0 m/s 36.9° Θ V