Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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].
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].
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.