What traumatic event does Zitkala-Sa describe at the school?

The forcible cutting of her long hair

Why was the cutting of her hair so humiliating to Zitkala-Sa?

Among her people, shingled hair was worn by cowards and mourners

Define the scalar (dot) product of two vectors.

\(\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta\), where \(\theta\) is the angle between them; it is a scalar.

Define the vector (cross) product.

\(\vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\sin\theta\,\hat{n}\), a vector perpendicular to both \(\vec{a}\) and \(\vec{b}\).

When are two vectors perpendicular?

When their dot product is zero: \(\vec{a}\cdot\vec{b}=0\) (with neither vector zero).

When are two vectors parallel?

When their cross product is the zero vector: \(\vec{a}\times\vec{b}=\vec{0}\).

How do you find a unit vector along \(\vec{a}\)?

\(\hat{a}=\dfrac{\vec{a}}{|\vec{a}|}\), dividing the vector by its magnitude.

What does \(|\vec{a}\times\vec{b}|\) represent geometrically?

It equals the area of the parallelogram with adjacent sides \(\vec{a}\) and \(\vec{b}\).

Write the magnitude of \(\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\).

\(|\vec{a}|=\sqrt{a_1^2+a_2^2+a_3^2}\).

State the section formula for internal division.

The point dividing \(\vec{a}\) and \(\vec{b}\) in ratio \(m:n\) internally has position vector \(\dfrac{m\vec{b}+n\vec{a}}{m+n}\).

What are direction cosines of a vector?

If a vector makes angles \(\alpha,\beta,\gamma\) with the axes, its direction cosines are \(\cos\alpha,\cos\beta,\cos\gamma\), satisfying \(\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1\).

How is the angle between two vectors found?

From \(\cos\theta=\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}\).

Vector Algebra

Class 12 Mathematics · Mathematics Part II · Chapter 4

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CBSE Class 12 · Mathematics

Integrals: 35 marks across the four papers

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CBSE Class 12 · Mathematics

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Summary

A vector has both magnitude and direction, in contrast to a scalar. This chapter introduces position vectors, direction cosines and direction ratios, and the representation \(\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\). Vectors are added by the triangle and parallelogram laws and scaled by real numbers. Two products are central: the scalar (dot) product \(\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta\), which measures projection and tests perpendicularity, and the vector (cross) product \(\vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\sin\theta\,\hat{n}\), whose magnitude gives the area of the parallelogram spanned by the vectors and whose direction is perpendicular to both. The unit vector \(\hat{a}=\dfrac{\vec{a}}{|\vec{a}|}\) and the section formula are also covered. These tools underpin three-dimensional geometry and physics.

Vectors and scalarsPosition vectors and direction cosinesAddition and scalar multiplicationScalar (dot) productVector (cross) product

Key terms

Vector

A quantity with both magnitude and direction.

Position vector

The vector from the origin to a point.

Unit vector

\(\hat{a}=\dfrac{\vec{a}}{|\vec{a}|}\), a vector of magnitude one.

Dot product

\(\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta\), a scalar.

Cross product

\(\vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\sin\theta\,\hat{n}\), a vector.

Direction cosines

The cosines of the angles a vector makes with the axes.

Extra questions & answers

Try one from this chapter

A vector is a quantity having:

Flashcards & interactive practice