Which of the following vectors is perpendicular to $3old{i} + 2old{j} - 5old{k}$?
A - HSC - SSCE Mathematics Extension 2 - Question 1 - 2024 - Paper 1
Question 1
Which of the following vectors is perpendicular to $3old{i} + 2old{j} - 5old{k}$?
A. $-old{i} - old{j} + old{k}$
B. $old{i} + old{j} - old{k}$
C. $-2old{i... show full transcript
Worked Solution & Example Answer:Which of the following vectors is perpendicular to $3old{i} + 2old{j} - 5old{k}$?
A - HSC - SSCE Mathematics Extension 2 - Question 1 - 2024 - Paper 1
Step 1
Determine the vector components
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Answer
The given vector is represented as:
old{v} = 3old{i} + 2old{j} - 5old{k}
We need to check which of the given vectors is perpendicular to this vector.
Step 2
Check the first option: $-old{i} - old{j} + old{k}$
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Answer
Calculate the dot product:
\
\bold{v} ullet old{v_1} = (3)(-1) + (2)(-1) + (-5)(1) = -3 - 2 - 5 = -10$$
This is not zero, so the vector is not perpendicular.
Step 3
Check the second option: $\bold{i} + \bold{j} - \bold{k}$
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Answer
Calculate the dot product:
\
\bold{v} ullet old{v_2} = (3)(1) + (2)(1) + (-5)(-1) = 3 + 2 + 5 = 10$$
This is not zero, so the vector is not perpendicular.
Step 4
Check the third option: $-2\bold{i} + 3\bold{j} + \bold{k}$
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Answer
Calculate the dot product:
\
\bold{v} ullet old{v_3} = (3)(-2) + (2)(3) + (-5)(1) = -6 + 6 - 5 = -5$$
This is not zero, so the vector is not perpendicular.
Step 5
Check the fourth option: $3\bold{i} - 2\bold{j} + \bold{k}$
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Answer
Calculate the dot product:
\
\bold{v} ullet old{v_4} = (3)(3) + (2)(-2) + (-5)(1) = 9 - 4 - 5 = 0$$
This is zero, therefore the vector is perpendicular.
