Vector Planes (Edexcel A-Level Further Mathematics): Quizzes

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Equations of planes
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In the vector form r=a+λb+μc\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c}, what does r\mathbf{r} represent?

Position vector of any point on the plane

In Cartesian form ax+by+cz=dax + by + cz = d, what does (a,b,c)(a, b, c) represent?

Normal vector perpendicular to plane

What is the relationship between the normal vector n\mathbf{n} and direction vectors b\mathbf{b} and c\mathbf{c}?

n\mathbf{n} is perpendicular to both

How is the normal vector n\mathbf{n} calculated from direction vectors b\mathbf{b} and c\mathbf{c}?

n=b×c\mathbf{n} = \mathbf{b} \times \mathbf{c}

What is the formula for dd in ax+by+cz=dax + by + cz = d using point (x1,y1,z1)(x_1, y_1, z_1) on the plane?

d=ax1+by1+cz1d = ax_1 + by_1 + cz_1

What requirement must b\mathbf{b} and c\mathbf{c} satisfy in vector form r=a+λb+μc\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c}?

They must be non-parallel

Which vectors lie in the plane?

Direction vectors

What is the first step to convert vector form to Cartesian form?

Find the normal vector

To find a plane from three points, what is the first step?

Find two direction vectors

Which statement correctly distinguishes direction from normal vectors?

Direction in plane; normal perpendicular

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