What is the vector equation of the line through $A(3, -2, 1)$ and $B(1, 0, 4)$ ?

\vec{r} = (3\hat{i} - 2\hat{j} + \hat{k}) + \lambda(\hat{i} + 4\hat{k})

Correct answerYour answer

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

Correct answerYour answer

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

Correct answerYour answer

\vec{r} = (\hat{i} + 4\hat{k}) + \lambda(3\hat{i} - 2\hat{j} + \hat{k})

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

Its direction is $\vec{b} - \vec{a} = (\hat{i} + 4\hat{k}) - (3\hat{i} - 2\hat{j} + \hat{k}) = -2\hat{i} + 2\hat{j} + 3\hat{k}$ .
Through $A$ : $\vec{r} = (3\hat{i} - 2\hat{j} + \hat{k}) + \lambda(-2\hat{i} + 2\hat{j} + 3\hat{k})$ .

Worth noting

$\hat{i} + 4\hat{k}$ is the position vector of $B$ , not the direction from $A$ to $B$ .

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

What is the vector equation of the line through A(3,−2,1)A(3, -2, 1)A(3,−2,1) and B(1,0,4)B(1, 0, 4)B(1,0,4)?