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Three vectors can be expressed as follows: \( \vec{FG} = -2\hat{i} - 6\hat{j} - 3\hat{k} \) \( \vec{GH} = 3\hat{i} + 9\hat{j} - 7\hat{k} \) \( \vec{EH} = 2\hat{i} + 3\hat{j} + \hat{k} \) (a) Find \( \vec{FH} \) - Scottish Highers Maths - Question 7 - 2016
Question 7
Three vectors can be expressed as follows: \( \vec{FG} = -2\hat{i} - 6\hat{j} - 3\hat{k} \) \( \vec{GH} = 3\hat{i} + 9\hat{j} - 7\hat{k} \) \( \vec{EH} = 2\hat{i}... show full transcript
Worked Solution & Example Answer:Three vectors can be expressed as follows: \( \vec{FG} = -2\hat{i} - 6\hat{j} - 3\hat{k} \) \( \vec{GH} = 3\hat{i} + 9\hat{j} - 7\hat{k} \) \( \vec{EH} = 2\hat{i} + 3\hat{j} + \hat{k} \) (a) Find \( \vec{FH} \) - Scottish Highers Maths - Question 7 - 2016
Step 1
Find \( \vec{FH} \).
Answer
To find ( \vec{FH} ), we use the relation: [ \vec{FH} = \vec{FG} + \vec{GH} ] Substituting the given vectors: [ \vec{FH} = (-2\hat{i} - 6\hat{j} - 3\hat{k}) + (3\hat{i} + 9\hat{j} - 7\hat{k}) ] Combining the components: [ \vec{FH} = (-2 + 3)\hat{i} + (-6 + 9)\hat{j} + (-3 - 7)\hat{k} ] [ \vec{FH} = 1\hat{i} + 3\hat{j} - 10\hat{k} ]
Step 2
Hence, or otherwise, find \( \vec{FE} \).
Answer
To find ( \vec{FE} ), we can express it using the relation: [ \vec{FE} = \vec{FH} + \vec{HE} ] Where ( \vec{HE} = \vec{EH} - \vec{GH} ). Thus, substituting: [ \vec{FE} = \vec{FH} + (\vec{EH} - \vec{GH}) ] Substituting the values: [ \vec{FE} = (1\hat{i} + 3\hat{j} - 10\hat{k}) + ((2\hat{i} + 3\hat{j} + \hat{k}) - (3\hat{i} + 9\hat{j} - 7\hat{k})) ] Calculating ( \vec{HE} ): [ \vec{HE} = (2 - 3)\hat{i} + (3 - 9)\hat{j} + (1 + 7)\hat{k} = -1\hat{i} - 6\hat{j} + 8\hat{k} ] Finally, substituting: [ \vec{FE} = (1 - 1)\hat{i} + (3 - 6)\hat{j} + (-10 + 8)\hat{k} ] [ \vec{FE} = 0\hat{i} - 3\hat{j} - 2\hat{k} ] Hence: [ \vec{FE} = -1\hat{i} - 5\hat{j} ]