What is the vector equation of the line through the point $(2, -1, 4)$ parallel to $\hat{i} + \hat{j} - 2\hat{k}$ ?

\vec{r} = (\hat{i} + \hat{j} - 2\hat{k}) + \lambda(2\hat{i} - \hat{j} + 4\hat{k})

Correct answerYour answer

\vec{r} = (2\hat{i} - \hat{j} + 4\hat{k}) + \lambda(\hat{i} + \hat{j} - 2\hat{k})

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\vec{r} = \lambda(3\hat{i} + 2\hat{k})

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\vec{r} = (2\hat{i} - \hat{j} + 4\hat{k}) \times (\hat{i} + \hat{j} - 2\hat{k})

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Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

A line through $\vec{a}$ parallel to $\vec{b}$ is $\vec{r} = \vec{a} + \lambda\vec{b}$ .
Here $\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k}$ and $\vec{b} = \hat{i} + \hat{j} - 2\hat{k}$ .

Worth noting

Swapping $\vec{a}$ and $\vec{b}$ gives a line through the wrong point, in the wrong direction.

Ten questions from Three-Dimensional Geometry, at your own pace.

What is the vector equation of the line through the point (2,−1,4)(2, -1, 4)(2,−1,4) parallel to i^+j^−2k^\hat{i} + \hat{j} - 2\hat{k}i^+j^​−2k^?