Vectors and their components.

Step 1 of 6

Components

A vector is a quantity with a direction, drawn as an arrow: a walk of 10 m to the north-east, a velocity, a force. Two numbers fix the arrow $u$⁠: its length, and the angle it makes with the horizontal axis.

The same arrow can also be described as two steps: a horizontal step, $u_x$⁠, and a vertical step, $u_y$⁠. These are the vector’s components. The arrow, its horizontal component and its vertical component make a right triangle with the arrow as its hypotenuse, so $u_x = |u| \cos\theta$$u_x =$$|u| \cos\theta$⁠ and $u_y = |u| \sin\theta$$u_y =$$|u| \sin\theta$⁠. Answer, then choose Worked out under Look inside.

Step 2 of 6

A negative component

When the arrow points left, its horizontal component is negative, and when it points down, its vertical component is negative, like the cosine and sine on the unit circle. The formula $u_x = |u| \cos\theta$$u_x =$$|u| \cos\theta$⁠ takes care of the sign by itself. Here Length of u is locked; turn Direction of u.

Step 3 of 6

From components to length

The other way round: the components are known, $u = (3, 4)$⁠, and the length is hidden. The components are the legs, and the arrow is the hypotenuse. Answer, then choose Worked out under Look inside.

Step 4 of 6

Adding vectors

You walk $4$⁠ steps east, then $3$⁠ north. Each walk is a vector, $u$⁠ and $v$⁠, and where you end up is their sum, $u + v$⁠: put the tail of $v$⁠ on the head of $u$⁠, and the sum is the arrow from the start to the end. Predict how far from the start you will be, then choose Worked out under Look inside.

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Component by component

When the arrows are not at right angles, the safe way is to add components: the horizontal ones together and the vertical ones together. Here $u = (6, 2)$⁠ and $v = (-2, 3)$⁠. The sum is written on the top line of the board. Work out its length with Pythagoras, then choose Worked out under Look inside.

Step 6 of 6

Vectors at right angles

$u$⁠ is $4$⁠ steps east, and $v$⁠ is $3$⁠ steps in a direction you choose. When $v$⁠ points east, like $u$⁠, the lengths add: $4 + 3 = 7$⁠. Turn Direction of v and watch the length of $u + v$⁠.