Vectors - Maharashtra Board STD 12 Science Maths Questions

Q 1Answer the following questions in short.1 Mark

The direction ratios of the line which is perpendicular to the lines with direction ratios $-1,2,2$ and $0,2,1$ are

A
$-2,-1,-2$
B
$2,1,2$
C
$2,-1,-2$
D
$-2,1,-2$

Q 2Solve the Following Question.(2 Marks)2 Marks

Show that the points $A (3,2,-4), B (9,8,-10)$ and $C (-2,-3,1)$ are collinear.

Q 3Solve the Following Question.(2 Marks)2 Marks

If two non-zero vectors, $\bar{a}$ and $\bar{b}$ are perpendicular to each other, then $(\bar{a}-\bar{b})^2=\ldots \ldots$

A
$a^2-b^2$
B
$(\bar{a}+\bar{b})^2$
C
$\bar{a} \cdot \bar{b}$
D
$\bar{a} \times \bar{b}$

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Q 4Solve the Following Question.(2 Marks)2 Marks

If two non-zero vectors $\bar{a}$ and $\bar{b}$ are collinear then prove that there exist scalars $m$ and $n$ such that $m \bar{a}+n \bar{b}=\overline{0}$ and $(m, n) \neq(0,0)$.

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Q 5Solve the Following Question.(2 Marks)2 Marks

Prove that the volume of a parallelepiped whose coterminous edges as $\bar{a}, \bar{b}, \bar{c}$ is $[\bar{a} \bar{b} \bar{c}]$. Hence find the volume of a parallelepiped whose coterminous edges $\bar{i}+\bar{j} \cdot \bar{j}+\bar{k}$ and $\bar{k}+\bar{i}$

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Q 6Solve the Following Question.(2 Marks)2 Marks

If $A, B, C, D$ are four non-collinear points in the plane such that $\overline{A D}+\overline{B D}+\overline{C D}=\overline{0}$, then prove that point $D$ is the centroid of the $\triangle ABC$.

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Q 7Solve the Following Question.(3 Marks)3 Marks

Find the area of the parallelogram whose adjacent sides are the vectors $2 \hat{i}-2 \hat{j}+\hat{k}$ and $\hat{i}-3 \hat{j}-3 \hat{k}$.

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Q 8Solve the Following Question.(3 Marks)3 Marks

If the vectors $3 \hat{i}+5 \hat{k}, 4 \hat{i}+2 \hat{j}-3 \hat{k}$ and $3 \hat{i}+\hat{j}+4 \hat{k}$ are coterminus edges of a parallelopiped, then find the volume of the parallelopiped.

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Q 9Solve the Following Question.(3 Marks)3 Marks

Express $-\bar{i}-3 \bar{j}+4 \bar{k}$ as a linear combination of vectors $2 \bar{i}+\bar{j}-4 \bar{k}, 2 \bar{i}+3 \bar{k}$ and $3 \bar{i}+\bar{j}-2 \bar{k}$

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Q 10Solve the Following Question.(3 Marks)3 Marks

Using vector method, prove that the median of a triangle are concurrent.

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Q 11Solve the Following Question.(3 Marks)3 Marks

If $A (\bar{a})$ and $B (\bar{b})$ be any two points in the space and $R (\bar{r})$ be a point on the line segment $AB$ dividing it internally in the ratio $m: n$ then prove that $\bar{r}=\frac{m \bar{b}+n \bar{a}}{m+n}$

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Q 12Solve the Following Question.(4 Marks)4 Marks

Find the direction cosines of the line perpendicular to the line whose direction ratios are $-2,1,-1$ and $-3,-4,1$.

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Q 13Solve the Following Question.(4 Marks)4 Marks

Show that volume of parallelepiped with coterminous edges as $\bar{a}, \bar{b}$ and $\bar{c}$ is $[\bar{a} \bar{b} \bar{c}]$. hence find the volume of the parallelepiped whose coterminous are $\hat{i}+\hat{j}, \hat{j}+\hat{k}, \hat{k}+\hat{i}$

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Q 14Solve the Following Question.(4 Marks)4 Marks

Find the volume of a tetrahedron whose vertices are A(-1, 2,3), B(3,-2,1), C(2, 1,3) and D(-1,-2,4).

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Q 15Solve the Following Question.(4 Marks)4 Marks

Using vectors prove that the altitudes of a triangle are concurrent.

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Q 16Solve the Following Question.(4 Marks)4 Marks

Using vector method, find the incentre of the triangle whose vertices are $P(0,4,0), Q(0,0,3)$ and $R(0,4,3)$.

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