Vowels /æ/, /ʌ/, /a:/

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The vowels /æ/, /ʌ/, /a:/ have common used names.
/æ/ Near-Open Front Unrounded Vowel
/ʌ/ Open Mid-Back Unrounded Vowel
/а:/ Open Back Unrounded Vowel
There is a common problem for students who mispronounce the vowels /æ/, /ʌ/, /a:/. This is because many languages don't have them. Students often use one /a/ for all three sounds.

The letter æ was used in Old English to represent the vowel that's pronounced in Modern English. It comes from Latin. In the original Latin it's pronounced as /ai/.
/æ/ sound is produced with the mouth wide open and the tongue lying flat at the bottom, with its tip touching over the lower teeth, the lips are spread, almost as in a smile. How to do it? Open your mouth, spread your lips and put your tongue at the front, touching slightly over the lower teeth. Train your speaking apparatus by pronouncing the next words and phrases.
bad /bæd/
map /mæp/
add /æd/
plan /plæn/
lamp /læmp/
fat /fæt/
back /bæk/
can /kæn/
man /mæn/
hand /hænd/
fact /fækt/
crab /kræb/
catch /kætʃ/
trap /træp/
stamp /stæmp/
damp /dæmp/
lap /læp/
bank /bæŋk/
dad /dæd/
pad /pæd/
mad /mæd/
that /ðæt/
lad /læd/
cap /kæp/
hat /hæt/
black /blæk/
scat /skæt/
rank /ræŋk/
gas /ɡæs/
rack /ræk/
Harry and Pat stand hand in hand.
A very fat cat sat on a mat and rapidly ate a fat rat.
What a big black cat, a big black cat.
What a cat, what a cat!

/ʌ/ is more similar to the /a/. There are two small differences, though: 1. It's a bit shorter. 2. It's articulated with the mouth a little closer. The tongue lies down at the center of the mouth and the lips are neutral (neither spread, nor rounded). How to do it? Just say a short /a/ without opening too much your mouth. Leave your tongue at the center of your mouth. The back of the tongue rises to the front of the soft palate to half the distance. Let's try to practice.
but /bʌt/
love /lʌv/
blood /blʌd/
come /kʌm/
some /sʌm/
much /mʌtʆ/
such /sʌtʃ/
dull /dʌl/
gun /gʌn/
duck /dʌk/
luck /lʌk/
puck /pʌk/
does /dʌz/
cup /kʌp/
up /ʌp/
bus /bʌs/
bun /bʌn/
cut /kʌt/
fun /fʌn/
nut /nʌt/
hut /hʌt/
sum /sʌm/
numb /nʌm/
thumb /θʌm/
dumb /dʌm/
crumb /krʌm/
nun /nʌn/
done /dʌn/
son /sʌn/
A runner run a bus.
A funny puppy runs in front of a bus.
It makes a trouble for us.

/ɑː/ is a long sound produced with the mouth open and the tongue at the back of the mouth. How to do it? Place your tongue at the back and lay maximum flat, with your mouth wide open. Imagine a ENT doctor (ear, nose, throat specialist) check your throat when you train to pronounce /a:/. It's time to practice.
car /kɑː/
star /stɑː/
far /fɑː/
park /pɑːk/
dark /dɑːk/
class /klɑːs/
dance /dɑːns/
ask /ɑːsk/
task /tɑːsk/
fast /fɑːst/
last /lɑːst/
half /hɑːf/
bath /bɑːθ/
part /pɑːt/
hard /hɑːd/
glass /ɡlɑːs/
grass /ɡrɑːs/
chance /tʃɑːns/
aunt /ɑːnt/
grasp /ɡrɑːsp/
art /ɑːt/
calm /kɑːm/
laugh /lɑːf/
large /lɑːdʒ/
pass /pɑːs/
arm /ɑːm/
bar /bɑːr/
smart /smɑːt/

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Visualization of English sounds here.
https://soundsofspeech.uiowa.edu/
https://soundsofspeech.uiowa.edu/assets/images/sos-video.mp4