Vector Algebra

Fill in all the gaps, then press "Check" to check your answers. Use the "Hint" button to get a free letter if an answer is giving you trouble. You can also click on the "[?]" button to get a clue. Note that you will lose points if you ask for hints or clues!

1. If D, E, F are the mid-points of the sides BC, CA and AB respectively of a triangle ABC, then the value of

\vec{A D} + \vec{B E} + \vec{C F}

is .
2. If a vector makes angles

α, β, γ

with OX, OY and OZ respectively, then the value of

\sin^{2} α + \sin^{2} β + \sin^{2} γ

is .
3. The projection of the vector

\hat{i} + 3 \hat{j} + 7 \hat{k}

on the vector

2 \hat{i} - 3 \hat{j} + 6 \hat{k}

is .
4. If

\vec{a}

and

\vec{b}

are perpendicular vectors,

|\vec{a} + \vec{b}| = 3

and

|\vec{a}| = 5

, then the value of

|\vec{b}|

is .
5. The value of

\hat{i} . (\hat{j} \times \hat{k}) + \hat{j} . (\hat{i} \times \hat{k}) + \hat{k} . (\hat{i} \times \hat{j})

is .
6. The value of the area of the parallelogram determined by the vectors

2 \hat{i}

and

3 \hat{j}

is square units.
7. The value of

[2 \hat{i} 3 \hat{j} 4 \hat{k}]

is .
8. For any two vectors

\vec{a}

and

\vec{b}

of magnitudes 3 and 4 respectively, the value of

[\vec{a} \vec{b} \vec{a} \times \vec{b}] + {(\vec{a} . \vec{b})}^{2}

is .
9. The distance of the point (2, 3, 4) from the x-axis is .
10. A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7) parallel to the coordinate planes. The length of a diagonal of the parallelopiped is .

Vector Algebra - I

Gap-fill exercise