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Question 1
Find $a \cdot b$ where $a = 3 - 2 5$ and $b = - 1 42$ .
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $- 1$
Step-by-step walkthrough
Choose a solution method
Method #1Worked solution
Step 1: Recall the formula
For vectors in component form, $a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$ .
Step 2: Substitute components
$a \cdot b = 3 (- 1) + (- 2) (4) + 5 (2)$ .
Step 3: Compute each term
$3 (- 1) = - 3$ , $(- 2) (4) = - 8$ , $5 (2) = 10$ .
Step 4: Sum the terms
$- 3 - 8 + 10 = - 1$ .
Method #2Why the others are wrong
Step 1: Option B
$5$ comes from only summing $a_{2} b_{2} + a_{3} b_{3} = - 8 + 10 = 2$ incorrectly or mixing up terms; it does not include the correct first term contribution properly.
Step 2: Option C
$21$ results from adding magnitudes of products without the negative sign, e.g. $3 + 8 + 10$ .
Step 3: Option D
$1$ arises from a sign slip such as $- 3 - 8 + 10$ computed as $- 3 + 8 + 10$ then miscombined.
Step 4: Confirm correct value
Careful term-by-term calculation gives $- 1$ , which is option A.
Question 2
A vector is defined as $v = 4 - 3 0$ . Using the property $v \cdot v = ∣ v ∣^{2}$ , find $∣ v ∣$ .
No clue? Show me the answer
Correct answerCorrect!Incorrect
B $5$
Step-by-step walkthrough
Choose a solution method
Method #1Worked solution
Step 1: Recall the self-dot property
$v \cdot v = v_{1}^{2} + v_{2}^{2} + v_{3}^{2} = ∣ v ∣^{2}$ .
Step 2: Substitute components
$v \cdot v = 4^{2} + (- 3)^{2} + 0^{2} = 16 + 9 + 0 = 25$ .
Step 3: Take the square root
$∣ v ∣ = 25 = 5$ .
Step 4: State the result
The magnitude of $v$ is $5$ .
Method #2Why the others are wrong
Step 1: Option B
$7$ would come from incorrectly adding $∣4∣ + ∣ - 3 ∣ + 0 = 7$ instead of squaring and taking a square root.
Step 2: Option C
$1$ has no clear derivation from the correct components and is far too small.
Step 3: Option D
$25$ is $v \cdot v$ itself, forgetting to take the square root to get the magnitude.
Step 4: Confirm correct value
Only $5$ correctly applies both squaring and the square root.

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13 more questions in this topic

Free preview 2/15

Question 1
Find $a \cdot b$ where $a = 3 - 2 5$ and $b = - 1 42$ .
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $- 1$
Step-by-step walkthrough
Choose a solution method
Method #1Worked solution
Step 1: Recall the formula
For vectors in component form, $a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$ .
Step 2: Substitute components
$a \cdot b = 3 (- 1) + (- 2) (4) + 5 (2)$ .
Step 3: Compute each term
$3 (- 1) = - 3$ , $(- 2) (4) = - 8$ , $5 (2) = 10$ .
Step 4: Sum the terms
$- 3 - 8 + 10 = - 1$ .
Method #2Why the others are wrong
Step 1: Option B
$5$ comes from only summing $a_{2} b_{2} + a_{3} b_{3} = - 8 + 10 = 2$ incorrectly or mixing up terms; it does not include the correct first term contribution properly.
Step 2: Option C
$21$ results from adding magnitudes of products without the negative sign, e.g. $3 + 8 + 10$ .
Step 3: Option D
$1$ arises from a sign slip such as $- 3 - 8 + 10$ computed as $- 3 + 8 + 10$ then miscombined.
Step 4: Confirm correct value
Careful term-by-term calculation gives $- 1$ , which is option A.
Question 2
A vector is defined as $v = 4 - 3 0$ . Using the property $v \cdot v = ∣ v ∣^{2}$ , find $∣ v ∣$ .
No clue? Show me the answer
Correct answerCorrect!Incorrect
B $5$
Step-by-step walkthrough
Choose a solution method
Method #1Worked solution
Step 1: Recall the self-dot property
$v \cdot v = v_{1}^{2} + v_{2}^{2} + v_{3}^{2} = ∣ v ∣^{2}$ .
Step 2: Substitute components
$v \cdot v = 4^{2} + (- 3)^{2} + 0^{2} = 16 + 9 + 0 = 25$ .
Step 3: Take the square root
$∣ v ∣ = 25 = 5$ .
Step 4: State the result
The magnitude of $v$ is $5$ .
Method #2Why the others are wrong
Step 1: Option B
$7$ would come from incorrectly adding $∣4∣ + ∣ - 3 ∣ + 0 = 7$ instead of squaring and taking a square root.
Step 2: Option C
$1$ has no clear derivation from the correct components and is far too small.
Step 3: Option D
$25$ is $v \cdot v$ itself, forgetting to take the square root to get the magnitude.
Step 4: Confirm correct value
Only $5$ correctly applies both squaring and the square root.

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AHL 3.13—Scalar and vector products

Step-by-step walkthrough

Step 1: Recall the formula

Step 2: Substitute components

Step 3: Compute each term

Step 4: Sum the terms

Step 1: Option B

Step 2: Option C

Step 3: Option D

Step 4: Confirm correct value

Step-by-step walkthrough

Step 1: Recall the self-dot property

Step 2: Substitute components

Step 3: Take the square root

Step 4: State the result

Step 1: Option B

Step 2: Option C

Step 3: Option D

Step 4: Confirm correct value

13 more questions in this topic

AHL 3.13—Scalar and vector products

Step-by-step walkthrough

Step 1: Recall the formula

Step 2: Substitute components

Step 3: Compute each term

Step 4: Sum the terms

Step 1: Option B

Step 2: Option C

Step 3: Option D

Step 4: Confirm correct value

Step-by-step walkthrough

Step 1: Recall the self-dot property

Step 2: Substitute components

Step 3: Take the square root

Step 4: State the result

Step 1: Option B

Step 2: Option C

Step 3: Option D

Step 4: Confirm correct value

13 more questions in this topic