Sure! Let's solve these vector equations step-by-step.
Given vectors:
[tex]\[ a = 4i - 4j \][/tex]
[tex]\[ b = -5i + 8j \][/tex]
[tex]\[ c = -i - 3j \][/tex]
[tex]\[ d = 6i + 2j \][/tex]
### 1. Expression for [tex]\( 6i + 9j \)[/tex]
We need to find a combination of the given vectors [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] that results in [tex]\( 6i + 9j \)[/tex].
Consider the expression [tex]\( a + d + b \)[/tex]:
First, let's compute the components of [tex]\( a + d + b \)[/tex]:
[tex]\[
a + d + b = (4i - 4j) + (6i + 2j) + (-5i + 8j)
\][/tex]
Combining the [tex]\( i \)[/tex]-components:
[tex]\[
4i + 6i - 5i = 5i
\][/tex]
Combining the [tex]\( j \)[/tex]-components:
[tex]\[
-4j + 2j + 8j = 6j
\][/tex]
Thus, the resultant vector from [tex]\( a + d + b \)[/tex] is:
[tex]\[
5i + 6j
\][/tex]
We see that this vector does not match [tex]\( 6i + 9j \)[/tex].
### 2. Expression for [tex]\( 2j \)[/tex]
We need to find a combination of the given vectors [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] that results in [tex]\( 2j \)[/tex].
Consider the expression [tex]\( c + b \)[/tex]:
First, let's compute the components of [tex]\( c + b \)[/tex]:
[tex]\[
c + b = (-i - 3j) + (-5i + 8j)
\][/tex]
Combining the [tex]\( i \)[/tex]-components:
[tex]\[
-i - 5i = -6i
\][/tex]
Combining the [tex]\( j \)[/tex]-components:
[tex]\[
-3j + 8j = 5j
\][/tex]
Thus, the resultant vector from [tex]\( c + b \)[/tex] is:
[tex]\[
-6i + 5j
\][/tex]
We see that this vector does not match [tex]\( 2j \)[/tex].
### 3. Expression for [tex]\( -8i + 13j \)[/tex]
We need to find a combination of the given vectors [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] that results in [tex]\( -8i + 13j \)[/tex].
Consider the expression [tex]\( b + d + c \)[/tex]:
First, let's compute the components of [tex]\( b + d + c \)[/tex]:
[tex]\[
b + d + c = (-5i + 8j) + (6i + 2j) + (-i - 3j)
\][/tex]
Combining the [tex]\( i \)[/tex]-components:
[tex]\[
-5i + 6i - i = 0i
\][/tex]
Combining the [tex]\( j \)[/tex]-components:
[tex]\[
8j + 2j - 3j = 7j
\][/tex]
Thus, the resultant vector from [tex]\( b + d + c \)[/tex] is:
[tex]\[
0i + 7j
\][/tex]
We see that this vector does not match [tex]\( -8i + 13j \)[/tex].
### 4. Expression for [tex]\( -10i - 7j \)[/tex]
We need to find a combination of the given vectors [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] that results in [tex]\( -10i - 7j \)[/tex].
Consider the expression [tex]\( 2c + a \)[/tex]:
First, let's compute the components of [tex]\( 2c + a \)[/tex]:
[tex]\[
2c = 2(-i - 3j) = -2i - 6j
\][/tex]
Adding [tex]\( a \)[/tex]:
[tex]\[
2c + a = (-2i - 6j) + (4i - 4j)
\][/tex]
Combining the [tex]\( i \)[/tex]-components:
[tex]\[
-2i + 4i = 2i
\][/tex]
Combining the [tex]\( j \)[/tex]-components:
[tex]\[
-6j - 4j = -10j
\][/tex]
Thus, the resultant vector from [tex]\( 2c + a \)[/tex] is:
[tex]\[
2i - 10j
\][/tex]
We see that this vector does not match [tex]\( -10i - 7j \)[/tex].
