Answer :
To solve for \( u - v \) in component form, we need to follow these steps:
### Step 1: Find the components of vector \( u \)
1. Given: Initial point of \( u \) is \( (6, 8) \) and terminal point is \( (3, -2) \).
2. Calculate the components of vector \( u \):
[tex]\[ u_x = 3 - 6 = -3 \][/tex]
[tex]\[ u_y = -2 - 8 = -10 \][/tex]
So, vector \( u \) in component form is:
[tex]\[ u = \langle -3, -10 \rangle \][/tex]
### Step 2: Find the components of vector \( v \)
1. Given: Initial point of \( v \) is \( (-4, -3) \) and terminal point is \( (1, -7) \).
2. Calculate the components of vector \( v \):
[tex]\[ v_x = 1 - (-4) = 1 + 4 = 5 \][/tex]
[tex]\[ v_y = -7 - (-3) = -7 + 3 = -4 \][/tex]
So, vector \( v \) in component form is:
[tex]\[ v = \langle 5, -4 \rangle \][/tex]
### Step 3: Calculate \( u - v \) in component form
1. Components of \( u \) are \( \langle -3, -10 \rangle \) and components of \( v \) are \( \langle 5, -4 \rangle \).
2. Subtract the components of \( v \) from the components of \( u \):
[tex]\[ (u - v)_x = u_x - v_x = -3 - 5 = -8 \][/tex]
[tex]\[ (u - v)_y = u_y - v_y = -10 - (-4) = -10 + 4 = -6 \][/tex]
So, vector \( u - v \) in component form is:
[tex]\[ u - v = \langle -8, -6 \rangle \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\langle -8, -6 \rangle} \][/tex]
### Step 1: Find the components of vector \( u \)
1. Given: Initial point of \( u \) is \( (6, 8) \) and terminal point is \( (3, -2) \).
2. Calculate the components of vector \( u \):
[tex]\[ u_x = 3 - 6 = -3 \][/tex]
[tex]\[ u_y = -2 - 8 = -10 \][/tex]
So, vector \( u \) in component form is:
[tex]\[ u = \langle -3, -10 \rangle \][/tex]
### Step 2: Find the components of vector \( v \)
1. Given: Initial point of \( v \) is \( (-4, -3) \) and terminal point is \( (1, -7) \).
2. Calculate the components of vector \( v \):
[tex]\[ v_x = 1 - (-4) = 1 + 4 = 5 \][/tex]
[tex]\[ v_y = -7 - (-3) = -7 + 3 = -4 \][/tex]
So, vector \( v \) in component form is:
[tex]\[ v = \langle 5, -4 \rangle \][/tex]
### Step 3: Calculate \( u - v \) in component form
1. Components of \( u \) are \( \langle -3, -10 \rangle \) and components of \( v \) are \( \langle 5, -4 \rangle \).
2. Subtract the components of \( v \) from the components of \( u \):
[tex]\[ (u - v)_x = u_x - v_x = -3 - 5 = -8 \][/tex]
[tex]\[ (u - v)_y = u_y - v_y = -10 - (-4) = -10 + 4 = -6 \][/tex]
So, vector \( u - v \) in component form is:
[tex]\[ u - v = \langle -8, -6 \rangle \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\langle -8, -6 \rangle} \][/tex]
