Vector Addition Describe how to add vectors graphically. How do you resolve a vector into components? How are the components used to accomplish vector addition? How do you change components to polar form? How is vector subtraction accomplished? How is vector multiplication accomplished? Describe the differences between scalar and vector products of vectors. Source: http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Scalars and vectors Adding Vectors Vectors that are parallel can be added together Example 10 5 Sum is 15 Vector sum goes from tail of first vector to head of last vector being added together

Vector Addition Adding Vectors Vectors that are parallel but in opposite directions can be added together (direction of larger magnitude “wins”) Example +10 Add vector -5 Sum is +5 Vector sum goes from tail of first vector to head of last vector being added together

Vector Addition Adding Vectors Vectors at right angles can be added head to tail (tip to tail) Vector sum found using Pythagorean Theorem (a2 + b2 = c2) Vector Addition Vector Addition – Order does not matter Vector sum goes from tail of first vector to head of last vector being added together

Vector Addition Adding Vectors Vectors not parallel or at right angles may be added Using scale drawings of vectors added head to tail Using trigonometry

Vector Addition Adding Vectors - using scale drawings Vectors in different directions drawn to scale can be added head-to-tail Vector sum goes from tail of first vector to head of last vector being added together

Vector Addition Adding Vectors - using trigonometry Vector A at angle Θ has magnitude (size) A, direction Θ. Vector A equivalent to vector sum of Ax and Ay. Ax called x component; runs along (or parallel to) x axis Ay called y component; runs along (or parallel to) y axis

Vector Addition Adding Vectors - using trigonometry Notes: Remember that you can move a vector as long as you do not change its magnitude or direction. This may be necessary to find the opposite or adjacent side.

Vector Addition Adding Vectors - using trigonometry Case 1: (Ax is the side adjacent to angle Θ) Ax = cos(θ) x A, since Ax is the side adjacent to angle θ, A is the hypotenuse and cos = adj/hyp Ay = sin(θ) x A, since Ay is (parallel to) the side opposite angle θ, A is the hypotenuse and sin = opp/hyp Note: Depending on the quadrant, you may have to add a – sign in front of Ax and/or Ay

Vector Addition Adding Vectors - using trigonometry Case 2: (Ay is the side adjacent to angle Θ) Ax = sin(θ) x A, since Ax is (parallel to) the side opposite to angle θ, A is the hypotenuse and sin = opp/hyp Ay = cos(θ) x A, since Ay is (parallel to) the side adjacent to angle θ, A is the hypotenuse and cos = adj/hyp Note: Depending on the quadrant, you may have to add a – sign in front of Ax and/or Ay

Vector Addition Adding Vectors - using trigonometry To add two vectors, determine the x and y component vectors for each Then add the x components together to find the x component of the vector sum And add the y components together to find the y component of the vector sum

Vector Addition Adding Vectors - using trigonometry Once you have the X and Y component of the vector sum, you can use the Pythagorean Theorem to find the hypotenuse And you can use arcsin() or arccos() with one of the components to find the angle

Vector Addition Vector Addition - Hyperphysics