Vectors and matrices — Question 4

QCAA 2024, Paper 1 · 1 mark

Q4 · 2024 · Technology-freeSimple familiar1 mark

QUESTION 4

A plane contains the point

(1,3,1)

and is normal to the vector

\hat{\mathbf i}+\hat{\mathbf j}+2\hat{\mathbf k}

. The vector equation of the plane is

(A)

\begin{pmatrix}x\\y\\z\end{pmatrix}\cdot\begin{pmatrix}1\\3\\1\end{pmatrix}=\begin{pmatrix}1\\1\\2\end{pmatrix}\cdot\begin{pmatrix}1\\3\\1\end{pmatrix}

(B)

\begin{pmatrix}x\\y\\z\end{pmatrix}\cdot\begin{pmatrix}1\\1\\2\end{pmatrix}=\begin{pmatrix}1\\3\\1\end{pmatrix}\cdot\begin{pmatrix}1\\1\\2\end{pmatrix}

(C)

\begin{pmatrix}x\\y\\z\end{pmatrix}\times\begin{pmatrix}1\\3\\1\end{pmatrix}=\begin{pmatrix}1\\1\\2\end{pmatrix}\times\begin{pmatrix}1\\3\\1\end{pmatrix}

(D)

\begin{pmatrix}x\\y\\z\end{pmatrix}\times\begin{pmatrix}1\\1\\2\end{pmatrix}=\begin{pmatrix}1\\3\\1\end{pmatrix}\times\begin{pmatrix}1\\1\\2\end{pmatrix}

ABCD

WORKED SOLUTION

1 mark

ANSWER

Worked solution

A plane through position vector

\mathbf a

with normal

\mathbf n

satisfies

\mathbf r\cdot\mathbf n=\mathbf a\cdot\mathbf n

. Here

\mathbf a=(1,3,1)

and

\mathbf n=(1,1,2)

, which is option B.

Selects option B.

[1 mark]

Worked explanation by QCE Vault; correct option verified against the QCAA answer key.